IndisputableMonolith.Gravity.MasterTheoremHandoffIntegration

   1import Mathlib
   2import IndisputableMonolith.Cosmology.DarkEnergyWofZStructural
   3import IndisputableMonolith.Gravity.MasterTheoremStructural
   4import IndisputableMonolith.Gravity.PageCurveDynamical
   5import IndisputableMonolith.Gravity.FreudenthalAxisStencilCoeffCert
   6import IndisputableMonolith.Gravity.PhysicalSixTetCubicDirichletInstance
   7import IndisputableMonolith.Gravity.QuantumChannel.PhysicalChannelAmplitudeLinear
   8import IndisputableMonolith.Gravity.Track1BCPhysicalResidual
   9import IndisputableMonolith.Gravity.TensorShearSector
  10import IndisputableMonolith.Verification.Track6FalsifierSensitivity
  11
  12/-!
  13# Gravity Track 7: Fork Handoff Integration
  14
  15This module is the integration-lane receipt for the parallel fork handoffs:
  16
  17* Fork A: Track 1.B `1B-SCH` stationarity reduction at `N=5`.
  18* Fork B: Track 1.B-PHY / 1.C physical residual and Bianchi interface.
  19* Fork C: Track 2.C many-body / `PiTensorProduct` amplitude-linear lift.
  20* Fork D: Track 3.C discrete recognition-tick Page-capacity transfer.
  21* Fork E: Track 4.C dark-energy `w(z)` falsifier-band refinement.
  22* Fork F: Track 6 falsifier-sensitivity packaging.
  23
  24It does not upgrade the discovery claim.  It records exactly what the new
  25endpoints prove and keeps the remaining Track 1 displacement-class leaves as
  26the next dependency.
  27-/
  28
  29namespace IndisputableMonolith
  30namespace Gravity
  31namespace MasterTheoremHandoffIntegration
  32
  33open IndisputableMonolith.Cosmology.DarkEnergyWofZStructural
  34open IndisputableMonolith.Gravity.PageCurveDynamical
  35open IndisputableMonolith.Gravity.MasterTheoremStructural
  36open IndisputableMonolith.Gravity.FreudenthalAxisStencilCoeffCert
  37open IndisputableMonolith.Gravity.PhysicalSixTetCubicDirichletInstance
  38open IndisputableMonolith.Gravity.QuantumChannel.AmplitudeLinearForced
  39open IndisputableMonolith.Gravity.Track1BCPhysicalResidual
  40open IndisputableMonolith.Verification.Track6FalsifierSensitivity
  41
  42/-! ## §1. Endpoint propositions consumed by Track 7 -/
  43
  44/-- Fork C endpoint: a finite sitewise family of binary physical channel
  45responses induces an amplitude-linear response on the many-body
  46`PiTensorProduct` ledger, acts sitewise on pure tensors, and inherits the
  47local density-only collapse. -/
  48def Track2ManyBodyEndpoint : Prop :=
  49  ∀ {ι : Type} [Fintype ι] [DecidableEq ι]
  50    (R_J : ι → JointSubstrate →ₗ[ℂ] JointSubstrate)
  51    (R_C : ι → Signal8 → Signal8)
  52    (hPhys : ∀ i : ι, PhysicalChannelResponseOf (R_J i) (R_C i)),
  53      IsManyBodyAmplitudeLinear
  54        (manyBodyPhysicalChannelResponse R_J R_C hPhys) ∧
  55      (∀ φ : ι → Signal8,
  56        manyBodyPhysicalChannelResponse R_J R_C hPhys
  57          (PiTensorProduct.tprod ℂ φ) =
  58        PiTensorProduct.tprod ℂ (fun i => R_C i (φ i))) ∧
  59      (∀ _hDen : ∀ i : ι, IsDensityOnly (R_C i),
  60        ∀ i : ι, ∀ φ : Signal8, R_C i φ = 0)
  61
  62/-- Fork C endpoint theorem consumed by the integration lane. -/
  63theorem track2_many_body_endpoint_holds : Track2ManyBodyEndpoint :=
  64  T0T8_many_body_physical_channel_amplitude_linear_one_statement
  65
  66/-- Fork A endpoint: seven `N=5` displacement-class Schläfli leaves imply
  67the weighted-deficit stationarity target consumed by the nonlinear Hessian
  68route. -/
  69def Track1SchlaefliReductionEndpoint : Prop :=
  70  CanonicalPeriodicSecondSchlaefliTypedEdgeSevenDispTargetsAtN5 →
  71    CanonicalPeriodicWeightedDeficitDerivativeStationaryTargetAtN5
  72
  73/-- Fork A endpoint theorem consumed by the integration lane. -/
  74theorem track1_schlaefli_reduction_endpoint_holds :
  75    Track1SchlaefliReductionEndpoint :=
  76  canonicalPeriodicWeightedDeficitDerivativeStationaryTargetAtN5_of_sevenDisp
  77
  78/-- Agent A `disp0` endpoint: the axis displacement-class leaf is reduced to
  79the finite base-vertex cancellation target with no filtered `PeriodicEdge`
  80bookkeeping left in the caller. -/
  81def Track1Disp0BaseVertexReductionEndpoint : Prop :=
  82  CanonicalPeriodicSecondSchlaefliTypedEdgeDisp0BaseVertexTargetAtN5 →
  83    CanonicalPeriodicSecondSchlaefliTypedEdgeDispTarget
  84      5 5 5 (by decide) (by decide) (by decide) (0 : Fin 7)
  85
  86/-- Agent A `disp0` endpoint theorem consumed by the integration lane. -/
  87theorem track1_disp0_base_vertex_reduction_endpoint_holds :
  88    Track1Disp0BaseVertexReductionEndpoint :=
  89  canonicalPeriodicSecondSchlaefliTypedEdgeDisp0TargetAtN5_of_baseVertexTarget
  90
  91/-- Agent A stationarity endpoint: the base-vertex `disp0` target follows from
  92stationarity of the partial `disp0` weighted deficit-derivative sum. -/
  93def Track1Disp0StationaryReductionEndpoint : Prop :=
  94  CanonicalPeriodicDisp0WeightedDeficitDerivativeBaseStationaryTargetAtN5 →
  95    CanonicalPeriodicSecondSchlaefliTypedEdgeDisp0BaseVertexTargetAtN5
  96
  97/-- Agent A stationarity endpoint theorem consumed by the integration lane. -/
  98theorem track1_disp0_stationary_reduction_endpoint_holds :
  99    Track1Disp0StationaryReductionEndpoint :=
 100  CanonicalPeriodicSecondSchlaefliTypedEdgeDisp0BaseVertexTargetAtN5_of_stationary
 101
 102/-- Parametric `disp d` endpoint: the matching base-vertex Schläfli leaf at
 103`N=5` follows from stationarity of the partial `disp d` weighted
 104deficit-derivative sum. -/
 105def Track1DispStationaryReductionEndpoint : Prop :=
 106  ∀ d : Fin 7,
 107    CanonicalPeriodicDispWeightedDeficitDerivativeBaseStationaryTargetAtN5 d →
 108      CanonicalPeriodicSecondSchlaefliTypedEdgeDispBaseVertexTargetAtN5 d
 109
 110/-- Parametric `disp d` endpoint theorem consumed by the integration lane. -/
 111theorem track1_disp_stationary_reduction_endpoint_holds :
 112    Track1DispStationaryReductionEndpoint :=
 113  fun d => CanonicalPeriodicSecondSchlaefliTypedEdgeDispBaseVertexTargetAtN5_of_stationary d
 114
 115/-- Seven-displacement stationarity bundle endpoint: the seven `disp d`
 116stationarity claims imply the canonical `N=5` weighted-deficit stationarity
 117target consumed by the Track 1.B local-correspondence/Hessian machinery. -/
 118def Track1SevenStationarityEndpoint : Prop :=
 119  CanonicalPeriodicDispWeightedDeficitDerivativeSevenBaseStationaryTargetsAtN5 →
 120    CanonicalPeriodicWeightedDeficitDerivativeStationaryTargetAtN5
 121
 122/-- Seven-displacement stationarity bundle endpoint theorem consumed by Track 7. -/
 123theorem track1_seven_stationarity_endpoint_holds :
 124    Track1SevenStationarityEndpoint :=
 125  canonicalPeriodicWeightedDeficitDerivativeStationaryTargetAtN5_of_sevenStationarity
 126
 127/-- Uniform stationarity endpoint: one theorem quantified over `d : Fin 7`
 128packages into the seven-stationarity bundle.  This is the clean next target for
 129the `1B-SCH` agent. -/
 130def Track1ForallDispStationarityPackagingEndpoint : Prop :=
 131  (∀ d : Fin 7, CanonicalPeriodicDispWeightedDeficitDerivativeBaseStationaryTargetAtN5 d) →
 132    CanonicalPeriodicDispWeightedDeficitDerivativeSevenBaseStationaryTargetsAtN5
 133
 134/-- Uniform stationarity packaging endpoint theorem consumed by Track 7. -/
 135theorem track1_forall_disp_stationarity_packaging_endpoint_holds :
 136    Track1ForallDispStationarityPackagingEndpoint :=
 137  canonicalPeriodicDispWeightedDeficitDerivativeSevenBaseStationaryTargetsAtN5_of_forall
 138
 139/-- Session 563 direct uniform stationarity endpoint: a single theorem quantified
 140over `d : Fin 7` closes the canonical `N=5` weighted-deficit stationarity
 141target without first exposing the seven-field bundle to the caller. -/
 142def Track1ForallDispStationarityEndpoint : Prop :=
 143  (∀ d : Fin 7, CanonicalPeriodicDispWeightedDeficitDerivativeBaseStationaryTargetAtN5 d) →
 144    CanonicalPeriodicWeightedDeficitDerivativeStationaryTargetAtN5
 145
 146/-- Session 563 endpoint theorem consumed by Track 7. -/
 147theorem track1_forall_disp_stationarity_endpoint_holds :
 148    Track1ForallDispStationarityEndpoint :=
 149  canonicalPeriodicWeightedDeficitDerivativeStationaryTargetAtN5_of_forallDispStationarity
 150
 151/-- Session 563 audit count for the direct uniform-stationarity handoff endpoint. -/
 152def track1ForallDispStationarityEndpointProjectionCount : ℕ := 1
 153
 154theorem track1ForallDispStationarityEndpointProjectionCount_eq_one :
 155    track1ForallDispStationarityEndpointProjectionCount = 1 := rfl
 156
 157/-- Session 568 Track 7 endpoint: total stationarity plus displacement symmetry
 158closes the uniform seven-displacement stationarity target. -/
 159def Track1TotalSymmetryStationarityReductionEndpoint : Prop :=
 160  CanonicalPeriodicDispWeightedDeficitDerivativeBaseSumTotalStationaryTargetAtN5 →
 161    CanonicalPeriodicDispWeightedDeficitDerivativeBaseSumDispSymmetryTargetAtN5 →
 162      ∀ d : Fin 7, CanonicalPeriodicDispWeightedDeficitDerivativeBaseStationaryTargetAtN5 d
 163
 164/-- Session 568 endpoint theorem consumed by Track 7. -/
 165theorem track1_total_symmetry_stationarity_reduction_endpoint_holds :
 166    Track1TotalSymmetryStationarityReductionEndpoint :=
 167  canonicalPeriodicDispWeightedDeficitDerivativeBaseStationaryTargetAtN5_of_totalStationary_and_dispSymmetry
 168
 169/-- Session 568 audit count for the total-plus-symmetry Track 7 endpoint. -/
 170def track1TotalSymmetryStationarityReductionEndpointProjectionCount : ℕ := 1
 171
 172theorem track1TotalSymmetryStationarityReductionEndpointProjectionCount_eq_one :
 173    track1TotalSymmetryStationarityReductionEndpointProjectionCount = 1 := rfl
 174
 175/-- Session 572 Track 7 endpoint: total stationarity plus displacement symmetry
 176closes the canonical `N=5` weighted-deficit stationarity target directly. -/
 177def Track1TotalSymmetryStationarityEndpoint : Prop :=
 178  CanonicalPeriodicDispWeightedDeficitDerivativeBaseSumTotalStationaryTargetAtN5 →
 179    CanonicalPeriodicDispWeightedDeficitDerivativeBaseSumDispSymmetryTargetAtN5 →
 180      CanonicalPeriodicWeightedDeficitDerivativeStationaryTargetAtN5
 181
 182/-- Session 572 endpoint theorem consumed by Track 7. -/
 183theorem track1_total_symmetry_stationarity_endpoint_holds :
 184    Track1TotalSymmetryStationarityEndpoint :=
 185  canonicalPeriodicWeightedDeficitDerivativeStationaryTargetAtN5_of_totalStationary_and_dispSymmetry
 186
 187/-- Session 572 audit count for the direct total-plus-symmetry Track 7 endpoint. -/
 188def track1TotalSymmetryStationarityEndpointProjectionCount : ℕ := 1
 189
 190theorem track1TotalSymmetryStationarityEndpointProjectionCount_eq_one :
 191    track1TotalSymmetryStationarityEndpointProjectionCount = 1 := rfl
 192
 193/-- Direct Schläfli-along-line endpoint: the conformal Schläfli identity
 194`V(t) = 0` for all `t` closes the full `N=5` weighted-deficit stationarity
 195target, bypassing the seven per-displacement-class stationarity targets
 196entirely. This is the clearest next proof surface: prove
 197`CanonicalPeriodicConformalSchlaefliAlongLineTargetAtN5` (a consequence of
 198the classical Schläfli differential identity applied at every parameter and
 199summed over tetrahedra), and the full Track 1.B stationarity input follows. -/
 200def Track1ConformalSchlaefliEndpoint : Prop :=
 201  CanonicalPeriodicConformalSchlaefliAlongLineTargetAtN5 →
 202    CanonicalPeriodicWeightedDeficitDerivativeStationaryTarget
 203      5 5 5 (by decide) (by decide) (by decide)
 204
 205/-- Direct Schläfli-along-line endpoint theorem consumed by Track 7. -/
 206theorem track1_conformal_schlaefli_endpoint_holds :
 207    Track1ConformalSchlaefliEndpoint :=
 208  canonicalPeriodicWeightedDeficitDerivativeStationaryTargetAtN5_of_conformalSchlaefli
 209
 210/-- Localized Schläfli proof endpoint: the two concrete non-flat targets
 211(global expansion/reindexing plus local tetrahedral Schläfli at each parameter)
 212produce the canonical conformal Schläfli along-line target at `N=5`. -/
 213def Track1ConformalSchlaefliLocalExpansionEndpoint : Prop :=
 214  CanonicalPeriodicConformalSchlaefliAlongLineExpansionTargetAtN5 →
 215    CanonicalPeriodicLocalConformalSchlaefliAlongLineTargetAtN5 →
 216      CanonicalPeriodicConformalSchlaefliAlongLineTargetAtN5
 217
 218/-- Localized Schläfli proof endpoint theorem consumed by Track 7. -/
 219theorem track1_conformal_schlaefli_local_expansion_endpoint_holds :
 220    Track1ConformalSchlaefliLocalExpansionEndpoint :=
 221  CanonicalPeriodicConformalSchlaefliAlongLineTargetAtN5_of_expansion_and_local
 222
 223/-- Near-zero expansion endpoint: the global derivative/reindexing half of the
 224localized Schläfli route is now proved for the canonical `N=5` periodic
 225Freudenthal torus. -/
 226def Track1ConformalSchlaefliNearZeroExpansionEndpoint : Prop :=
 227  CanonicalPeriodicConformalSchlaefliNearZeroExpansionTargetAtN5
 228
 229/-- Near-zero expansion endpoint theorem consumed by Track 7. -/
 230theorem track1_conformal_schlaefli_near_zero_expansion_endpoint_holds :
 231    Track1ConformalSchlaefliNearZeroExpansionEndpoint :=
 232  canonicalPeriodicConformalSchlaefliNearZeroExpansionTargetAtN5
 233
 234/-- Local near-zero Schläfli reduction endpoint: the remaining local target is
 235reduced to the actual non-flat squared-edge chain rule plus the closed-form
 236Schläfli zero at the deformed squared-edge tuple. -/
 237def Track1ConformalSchlaefliNearZeroLocalReductionEndpoint : Prop :=
 238  CanonicalPeriodicLocalConformalSchlaefliAngleSqEdgeChainRuleNearZeroTargetAtN5 →
 239    CanonicalPeriodicLocalConformalSchlaefliClosedFormZeroNearZeroTargetAtN5 →
 240      CanonicalPeriodicWeightedDeficitDerivativeStationaryTarget
 241        5 5 5 (by decide) (by decide) (by decide)
 242
 243/-- Local near-zero reduction endpoint theorem consumed by Track 7. -/
 244theorem track1_conformal_schlaefli_near_zero_local_reduction_endpoint_holds :
 245    Track1ConformalSchlaefliNearZeroLocalReductionEndpoint :=
 246  canonicalPeriodicWeightedDeficitDerivativeStationaryTargetAtN5_of_sqEdgeChainRule_and_closedFormZero
 247
 248/-- Near-zero local chain-rule endpoint: the actual non-flat squared-edge chain
 249rule is proved for the canonical `N=5` periodic Freudenthal torus near the flat
 250point. -/
 251def Track1ConformalSchlaefliNearZeroChainRuleEndpoint : Prop :=
 252  CanonicalPeriodicLocalConformalSchlaefliAngleSqEdgeChainRuleNearZeroTargetAtN5
 253
 254/-- Near-zero local chain-rule endpoint theorem consumed by Track 7. -/
 255theorem track1_conformal_schlaefli_near_zero_chain_rule_endpoint_holds :
 256    Track1ConformalSchlaefliNearZeroChainRuleEndpoint :=
 257  canonicalPeriodicLocalConformalSchlaefliAngleSqEdgeChainRuleNearZeroTargetAtN5
 258
 259/-- Near-zero closed-form Schläfli-zero endpoint: the algebraic local
 260Schläfli cancellation is proved at the deformed squared-edge tuple. -/
 261def Track1ConformalSchlaefliNearZeroClosedFormEndpoint : Prop :=
 262  CanonicalPeriodicLocalConformalSchlaefliClosedFormZeroNearZeroTargetAtN5
 263
 264/-- Near-zero closed-form endpoint theorem consumed by Track 7. -/
 265theorem track1_conformal_schlaefli_near_zero_closed_form_endpoint_holds :
 266    Track1ConformalSchlaefliNearZeroClosedFormEndpoint :=
 267  canonicalPeriodicLocalConformalSchlaefliClosedFormZeroNearZeroTargetAtN5
 268
 269/-- Near-zero local Schläfli endpoint: both local tetrahedral inputs have been
 270closed, so the local near-zero Schläfli identity itself is theorem-grade. -/
 271def Track1ConformalSchlaefliNearZeroLocalEndpoint : Prop :=
 272  CanonicalPeriodicLocalConformalSchlaefliNearZeroTargetAtN5
 273
 274/-- Near-zero local Schläfli endpoint theorem consumed by Track 7. -/
 275theorem track1_conformal_schlaefli_near_zero_local_endpoint_holds :
 276    Track1ConformalSchlaefliNearZeroLocalEndpoint :=
 277  canonicalPeriodicLocalConformalSchlaefliNearZeroTargetAtN5
 278
 279/-- Near-zero Schläfli stationarity endpoint: the global near-zero expansion and
 280the local near-zero Schläfli identity now close the full canonical `N=5`
 281weighted-deficit stationarity target. -/
 282def Track1ConformalSchlaefliNearZeroStationarityEndpoint : Prop :=
 283  CanonicalPeriodicWeightedDeficitDerivativeStationaryTarget
 284    5 5 5 (by decide) (by decide) (by decide)
 285
 286/-- Near-zero Schläfli stationarity endpoint theorem consumed by Track 7. -/
 287theorem track1_conformal_schlaefli_near_zero_stationarity_endpoint_holds :
 288    Track1ConformalSchlaefliNearZeroStationarityEndpoint :=
 289  canonicalPeriodicWeightedDeficitDerivativeStationaryTargetAtN5_from_nearZeroSchlaefli
 290
 291/-- Track 1.B packaging endpoint after the Schläfli closure: the canonical
 292`N=5` local Regge/J-cost correspondence now depends only on the mixed
 293hinge-deficit length-chain identity. -/
 294def Track1LocalCorrespondenceReducedToMixedLengthEndpoint : Prop :=
 295  CanonicalPeriodicMixedHingeDeficitLengthChainTargetAtN5 →
 296    CanonicalPeriodicEdgeStencilLocalCorrespondenceAtN5
 297
 298/-- Track 1.B packaging endpoint theorem consumed by Track 7. -/
 299theorem track1_local_correspondence_reduced_to_mixed_length_endpoint_holds :
 300    Track1LocalCorrespondenceReducedToMixedLengthEndpoint :=
 301  canonicalPeriodicEdgeStencilLocalCorrespondenceAtN5_of_mixedLengthChain
 302
 303/-- Session 202 Track 1.B audit endpoint: the old mixed length-chain
 304edge-stencil RHS is scalar-inconsistent with the exact finite `N=5`
 305single-vertex audit.  The replacement target is
 306`CanonicalPeriodicMixedHingeDeficitAxisStencilTargetAtN5`. -/
 307def Track1MixedLengthAuditObstructionEndpoint : Prop :=
 308  (12 : ℝ) ≠ 6 + 6 * Real.sqrt 2 + 2 * Real.sqrt 3
 309
 310/-- Session 202 audit endpoint theorem consumed by Track 7. -/
 311theorem track1_mixed_length_audit_obstruction_endpoint_holds :
 312    Track1MixedLengthAuditObstructionEndpoint :=
 313  canonicalPeriodicMixedLengthSingleVertexAudit_scalar_mismatch
 314
 315/-- Session 204 Track 1.B corrected-target endpoint: the corrected axis-stencil
 316mixed target follows from the global explicit-fiber axis-stencil identity. -/
 317def Track1MixedAxisStencilReductionEndpoint : Prop :=
 318  CanonicalPeriodicMixedHingeDeficitExplicitFiberAxisStencilTargetAtN5 →
 319    CanonicalPeriodicMixedHingeDeficitAxisStencilTargetAtN5
 320
 321/-- Session 204 corrected-target reduction theorem consumed by Track 7. -/
 322theorem track1_mixed_axis_stencil_reduction_endpoint_holds :
 323    Track1MixedAxisStencilReductionEndpoint :=
 324  canonicalPeriodicMixedHingeDeficitAxisStencilTargetAtN5_of_explicitFiberAxis
 325
 326/-- Session 207 Track 1.B exact rational coefficient endpoint: the
 327translation-normalized origin-offset residual coefficients for the corrected
 328`N=5` axis-stencil target vanish in Lean.  The full unordered-pair table is
 329still audited by the Python exact checker. -/
 330def Track1MixedAxisCoeffCertEndpoint : Prop :=
 331  originResidualCoeffsZero = true
 332
 333/-- Session 207 coefficient endpoint consumed by Track 7. -/
 334theorem track1_mixed_axis_coeff_cert_endpoint_holds :
 335    Track1MixedAxisCoeffCertEndpoint :=
 336  originResidualCoeffsZero_eq_true
 337
 338/-- Session 208 Track 1.B split-row probe: one non-origin row of the exact
 339`Rat` residual coefficient table also vanishes.  This validates the row-split
 340certificate route, while exposing that all 125 rows should be generated in a
 341sparser form or replaced by translation invariance. -/
 342def Track1MixedAxisRow100CoeffCertEndpoint : Prop :=
 343  rowResidualCoeffsZero (1, 0, 0) = true
 344
 345/-- Session 208 split-row coefficient endpoint consumed by Track 7. -/
 346theorem track1_mixed_axis_row100_coeff_cert_endpoint_holds :
 347    Track1MixedAxisRow100CoeffCertEndpoint :=
 348  rowResidualCoeffsZero_100_eq_true
 349
 350/-- Session 209 Track 1.B origin-row Prop endpoint: the boolean certificate is
 351now exposed as the theorem-shaped origin-row coefficient vanishing statement
 352used by the translation-invariance bridge. -/
 353def Track1MixedAxisOriginPropCoeffCertEndpoint : Prop :=
 354  ∀ v : Vertex5, mixedAxisResidualCoeff originVertex v = 0
 355
 356/-- Session 209 origin-row Prop endpoint consumed by Track 7. -/
 357theorem track1_mixed_axis_origin_prop_coeff_cert_endpoint_holds :
 358    Track1MixedAxisOriginPropCoeffCertEndpoint :=
 359  originResidualCoeffCert
 360
 361/-- Session 209 Track 1.B full-table reduction endpoint: translation invariance
 362of the exact rational residual coefficients upgrades the origin-row certificate
 363to the full `125 × 125` coefficient-vanishing certificate without compiling
 364all rows. -/
 365def Track1MixedAxisTranslationReductionEndpoint : Prop :=
 366  MixedAxisResidualCoeffTranslationInvariant → FullResidualCoeffCert
 367
 368/-- Session 209 translation-reduction endpoint consumed by Track 7. -/
 369theorem track1_mixed_axis_translation_reduction_endpoint_holds :
 370    Track1MixedAxisTranslationReductionEndpoint :=
 371  fullResidualCoeffCert_of_translationInvariant
 372
 373/-- Session 209 Track 1.B axis-stencil RHS translation endpoint: the corrected
 374three-axis stencil coefficient model is translation invariant on the `N=5`
 375periodic torus. -/
 376def Track1MixedAxisStencilRhsTranslationEndpoint : Prop :=
 377  AxisStencilResidualCoeffTranslationInvariant
 378
 379/-- Session 209 axis-stencil RHS translation endpoint consumed by Track 7. -/
 380theorem track1_mixed_axis_stencil_rhs_translation_endpoint_holds :
 381    Track1MixedAxisStencilRhsTranslationEndpoint :=
 382  axisStencilResidualCoeff_translationInvariant
 383
 384/-- Session 209 Track 1.B LHS-only reduction endpoint: after the RHS
 385translation theorem, the full coefficient certificate follows from the single
 386remaining mixed explicit-fiber LHS translation reindexing theorem. -/
 387def Track1MixedAxisLhsTranslationReductionEndpoint : Prop :=
 388  MixedAxisLhsCoeffTranslationInvariant → FullResidualCoeffCert
 389
 390/-- Session 209 LHS-only translation-reduction endpoint consumed by Track 7. -/
 391theorem track1_mixed_axis_lhs_translation_reduction_endpoint_holds :
 392    Track1MixedAxisLhsTranslationReductionEndpoint :=
 393  fullResidualCoeffCert_of_lhs_translationInvariant
 394
 395/-- Session 211 Track 1.B edge-summand reduction endpoint: the full corrected
 396coefficient certificate follows from translation invariance of one
 397`mixedAxisEdgeLhsCoeff` summand, because `translateEdge5Equiv` reindexes the
 398outer finite edge sum. -/
 399def Track1MixedAxisEdgeLhsTranslationReductionEndpoint : Prop :=
 400  MixedAxisEdgeLhsCoeffTranslationInvariant → FullResidualCoeffCert
 401
 402/-- Session 211 edge-summand translation-reduction endpoint consumed by Track 7. -/
 403theorem track1_mixed_axis_edge_lhs_translation_reduction_endpoint_holds :
 404    Track1MixedAxisEdgeLhsTranslationReductionEndpoint :=
 405  fullResidualCoeffCert_of_edge_lhs_translationInvariant
 406
 407/-- Session 212 Track 1.B local edge-summand translation endpoint: the mixed
 408explicit-fiber LHS edge contribution is invariant under `N=5` torus
 409translation. -/
 410def Track1MixedAxisEdgeLhsTranslationEndpoint : Prop :=
 411  MixedAxisEdgeLhsCoeffTranslationInvariant
 412
 413/-- Session 212 edge-summand translation endpoint consumed by Track 7. -/
 414theorem track1_mixed_axis_edge_lhs_translation_endpoint_holds :
 415    Track1MixedAxisEdgeLhsTranslationEndpoint :=
 416  mixedAxisEdgeLhsCoeff_translationInvariant
 417
 418/-- Session 212 Track 1.B LHS translation endpoint: the full mixed
 419explicit-fiber LHS coefficient model is translation invariant after reindexing
 420the periodic edge sum. -/
 421def Track1MixedAxisLhsTranslationEndpoint : Prop :=
 422  MixedAxisLhsCoeffTranslationInvariant
 423
 424/-- Session 212 mixed LHS translation endpoint consumed by Track 7. -/
 425theorem track1_mixed_axis_lhs_translation_endpoint_holds :
 426    Track1MixedAxisLhsTranslationEndpoint :=
 427  mixedAxisLhsCoeff_translationInvariant
 428
 429/-- Session 212 Track 1.B full rational residual certificate: every coefficient
 430in the corrected `N=5` axis-stencil residual vanishes. -/
 431def Track1MixedAxisFullResidualCoeffCertEndpoint : Prop :=
 432  FullResidualCoeffCert
 433
 434/-- Session 212 full residual coefficient endpoint consumed by Track 7. -/
 435theorem track1_mixed_axis_full_residual_coeff_cert_endpoint_holds :
 436    Track1MixedAxisFullResidualCoeffCertEndpoint :=
 437  fullResidualCoeffCert
 438
 439/-- Session 230 Track 1.B scalar finite RHS endpoint: the corrected three-axis
 440stencil is sound against the unordered coefficient expansion. -/
 441def Track1MixedAxisRhsSoundnessEndpoint : Prop :=
 442  AxisStencilCoeffSoundnessAtN5
 443
 444/-- Session 230 RHS soundness endpoint consumed by Track 7. -/
 445theorem track1_mixed_axis_rhs_soundness_endpoint_holds :
 446    Track1MixedAxisRhsSoundnessEndpoint :=
 447  axisStencilCoeffSoundnessAtN5
 448
 449/-- Session 230 Track 1.B scalar finite LHS endpoint: the real explicit-fiber
 450mixed LHS equals the rational unordered coefficient model. -/
 451def Track1MixedAxisExplicitFiberLhsSoundnessEndpoint : Prop :=
 452  ExplicitFiberMixedLhsCoeffSoundnessAtN5
 453
 454/-- Session 230 explicit-fiber LHS soundness endpoint consumed by Track 7. -/
 455theorem track1_mixed_axis_explicit_fiber_lhs_soundness_endpoint_holds :
 456    Track1MixedAxisExplicitFiberLhsSoundnessEndpoint :=
 457  explicitFiberMixedLhsCoeffSoundnessAtN5
 458
 459/-- Session 230 Track 1.B full scalar finite soundness endpoint: the real
 460explicit-fiber residual is the unordered rational residual expansion. -/
 461def Track1MixedAxisExplicitFiberAxisSoundnessEndpoint : Prop :=
 462  ExplicitFiberAxisStencilCoeffSoundnessAtN5
 463
 464/-- Session 230 full scalar finite soundness endpoint consumed by Track 7. -/
 465theorem track1_mixed_axis_explicit_fiber_axis_soundness_endpoint_holds :
 466    Track1MixedAxisExplicitFiberAxisSoundnessEndpoint :=
 467  explicitFiberAxisStencilCoeffSoundnessAtN5
 468
 469/-- Session 230 closed corrected explicit-fiber axis-stencil target at `N = 5`. -/
 470def Track1MixedAxisExplicitFiberAxisStencilTargetEndpoint : Prop :=
 471  CanonicalPeriodicMixedHingeDeficitExplicitFiberAxisStencilTargetAtN5
 472
 473/-- Session 230 explicit-fiber target endpoint consumed by Track 7. -/
 474theorem track1_mixed_axis_explicit_fiber_axis_stencil_target_endpoint_holds :
 475    Track1MixedAxisExplicitFiberAxisStencilTargetEndpoint :=
 476  canonicalPeriodicMixedHingeDeficitExplicitFiberAxisStencilTargetAtN5
 477
 478/-- Session 230 closed corrected mixed axis-stencil target at `N = 5`. -/
 479def Track1MixedAxisCorrectedAxisStencilTargetEndpoint : Prop :=
 480  CanonicalPeriodicMixedHingeDeficitAxisStencilTargetAtN5
 481
 482/-- Session 230 corrected axis-stencil endpoint consumed by Track 7. -/
 483theorem track1_mixed_axis_corrected_axis_stencil_target_endpoint_holds :
 484    Track1MixedAxisCorrectedAxisStencilTargetEndpoint :=
 485  canonicalPeriodicMixedHingeDeficitAxisStencilTargetAtN5
 486
 487/-- Session 213 Track 1.B packaging endpoint: once the real explicit-fiber
 488residual is identified with the rational coefficient model, the closed
 489coefficient certificate proves the corrected explicit-fiber axis-stencil target
 490at `N=5`. -/
 491def Track1MixedAxisCoeffSoundnessToExplicitFiberEndpoint : Prop :=
 492  ExplicitFiberAxisStencilCoeffSoundnessAtN5 →
 493    CanonicalPeriodicMixedHingeDeficitExplicitFiberAxisStencilTargetAtN5
 494
 495/-- Session 213 coefficient-soundness packaging endpoint consumed by Track 7. -/
 496theorem track1_mixed_axis_coeff_soundness_to_explicit_fiber_endpoint_holds :
 497    Track1MixedAxisCoeffSoundnessToExplicitFiberEndpoint :=
 498  canonicalPeriodicMixedHingeDeficitExplicitFiberAxisStencilTargetAtN5_of_coeffSoundness
 499
 500/-- Session 213 Track 1.B corrected-target packaging endpoint: the same
 501coefficient-soundness bridge also proves the corrected mixed axis-stencil target
 502through the Session 204 explicit-fiber wrapper. -/
 503def Track1MixedAxisCoeffSoundnessToAxisStencilEndpoint : Prop :=
 504  ExplicitFiberAxisStencilCoeffSoundnessAtN5 →
 505    CanonicalPeriodicMixedHingeDeficitAxisStencilTargetAtN5
 506
 507/-- Session 213 coefficient-soundness to corrected-axis endpoint consumed by Track 7. -/
 508theorem track1_mixed_axis_coeff_soundness_to_axis_stencil_endpoint_holds :
 509    Track1MixedAxisCoeffSoundnessToAxisStencilEndpoint :=
 510  canonicalPeriodicMixedHingeDeficitAxisStencilTargetAtN5_of_coeffSoundness
 511
 512/-- Session 215 Track 1.D endpoint: edge-level perturbations are the right
 513carrier for tensor/shear work, and nontrivial rectangle shear is not
 514vertex-conformal. -/
 515def Track1DTensorShearScaffoldIntegrationEndpoint : Prop :=
 516  TensorShearSector.Track1DTensorShearScaffoldEndpoint
 517
 518/-- Session 215 tensor/shear scaffold endpoint consumed by Track 7. -/
 519theorem track1D_tensor_shear_scaffold_integration_endpoint_holds :
 520    Track1DTensorShearScaffoldIntegrationEndpoint :=
 521  TensorShearSector.track1D_tensorShearScaffoldEndpoint_holds
 522
 523/-- Track 1.D endpoint: TT is now represented as finite
 524orthogonality to the periodic conformal slice and a caller-supplied gauge slice;
 525constructing the actual projectors remains the next tensor-sector proof. -/
 526def Track1DTTOrthogonalSurfaceEndpoint : Prop :=
 527  ∀ (GaugePotential : Type)
 528    (gaugeMap : GaugePotential → TensorShearSector.PeriodicEdgePerturbation5),
 529    TensorShearSector.PeriodicTTOrthogonal5 GaugePotential gaugeMap (fun _ => 0) ∧
 530    (TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5 GaugePotential gaugeMap →
 531      TensorShearSector.PeriodicFreudenthalTTDecompositionTargetAtN5
 532        TensorShearSector.PeriodicConformalLogSubspace5
 533        (TensorShearSector.PeriodicGaugeSubspace5 GaugePotential gaugeMap)
 534        (TensorShearSector.PeriodicTTOrthogonal5 GaugePotential gaugeMap))
 535
 536/-- Tensor/TT orthogonality surface endpoint consumed by Track 7. -/
 537theorem track1D_tt_orthogonal_surface_endpoint_holds :
 538    Track1DTTOrthogonalSurfaceEndpoint := by
 539  intro GaugePotential gaugeMap
 540  exact ⟨TensorShearSector.periodicTTOrthogonal5_zero GaugePotential gaugeMap,
 541    TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_to_target
 542      GaugePotential gaugeMap⟩
 543
 544/-- Track 1.D endpoint: concrete finite projector data is sufficient to close
 545the periodic Freudenthal conformal/gauge/TT decomposition target. -/
 546def Track1DTTProjectorDataReductionEndpoint : Prop :=
 547  ∀ (GaugePotential : Type)
 548    (gaugeMap : GaugePotential → TensorShearSector.PeriodicEdgePerturbation5),
 549    TensorShearSector.PeriodicTTProjectorData5 GaugePotential gaugeMap →
 550      TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
 551        GaugePotential gaugeMap
 552
 553/-- Projector-data reduction endpoint consumed by Track 7. -/
 554theorem track1D_tt_projector_data_reduction_endpoint_holds :
 555    Track1DTTProjectorDataReductionEndpoint := by
 556  intro GaugePotential gaugeMap D
 557  exact TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_projectorData D
 558
 559/-- Track 1.D endpoint: finite spanning-generator projector data is sufficient to
 560build the projector data and close the orthogonal decomposition target. -/
 561def Track1DTTFiniteGeneratorProjectorReductionEndpoint : Prop :=
 562  ∀ (GaugePotential CIdx GIdx : Type)
 563    (cFintype : Fintype CIdx) (gFintype : Fintype GIdx)
 564    (gaugeMap : GaugePotential → TensorShearSector.PeriodicEdgePerturbation5),
 565    @TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5
 566      GaugePotential gaugeMap CIdx GIdx cFintype gFintype →
 567      Nonempty (TensorShearSector.PeriodicTTProjectorData5 GaugePotential gaugeMap) ∧
 568      TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
 569        GaugePotential gaugeMap
 570
 571/-- Finite-generator projector-data reduction endpoint consumed by Track 7. -/
 572theorem track1D_tt_finite_generator_projector_reduction_endpoint_holds :
 573    Track1DTTFiniteGeneratorProjectorReductionEndpoint := by
 574  intro GaugePotential CIdx GIdx cFintype gFintype gaugeMap D
 575  letI := cFintype
 576  letI := gFintype
 577  exact ⟨⟨TensorShearSector.PeriodicTTProjectorData5.ofFiniteGeneratorData D⟩,
 578    TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_finiteGeneratorData D⟩
 579
 580/-- Track 1.D endpoint: the `N = 5` conformal slice already has an explicit
 581finite spanning family, indexed by encoded vertices. -/
 582def Track1DConformalGeneratorSpanEndpoint : Prop :=
 583  ∀ c : TensorShearSector.PeriodicEdgePerturbation5,
 584    TensorShearSector.PeriodicConformalLogSubspace5 c →
 585      ∃ coeff : Fin TensorShearSector.PeriodicTorus5.K.nV → ℝ,
 586        ∀ e, c e = ∑ v : Fin TensorShearSector.PeriodicTorus5.K.nV,
 587          coeff v * TensorShearSector.periodicConformalGenerator5 v e
 588
 589/-- Conformal-generator span endpoint consumed by Track 7. -/
 590theorem track1D_conformal_generator_span_endpoint_holds :
 591    Track1DConformalGeneratorSpanEndpoint :=
 592  TensorShearSector.periodicConformalLogSubspace5_spanned_by_encodedVertexGenerators
 593
 594/-- Track 1.D endpoint: after the conformal span is fixed by vertex generators,
 595it is enough to supply gauge-generator projector data. -/
 596def Track1DTTGaugeGeneratorProjectorReductionEndpoint : Prop :=
 597  ∀ (GaugePotential GIdx : Type)
 598    (gFintype : Fintype GIdx)
 599    (gaugeMap : GaugePotential → TensorShearSector.PeriodicEdgePerturbation5),
 600    @TensorShearSector.PeriodicTTGaugeGeneratorProjectorData5
 601      GaugePotential gaugeMap GIdx gFintype →
 602      Nonempty
 603        (@TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5
 604          GaugePotential gaugeMap
 605          (Fin TensorShearSector.PeriodicTorus5.K.nV) GIdx
 606          inferInstance gFintype) ∧
 607      Nonempty (TensorShearSector.PeriodicTTProjectorData5 GaugePotential gaugeMap) ∧
 608      TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
 609        GaugePotential gaugeMap
 610
 611/-- Gauge-generator projector-data reduction endpoint consumed by Track 7. -/
 612theorem track1D_tt_gauge_generator_projector_reduction_endpoint_holds :
 613    Track1DTTGaugeGeneratorProjectorReductionEndpoint := by
 614  intro GaugePotential GIdx gFintype gaugeMap D
 615  letI := gFintype
 616  let FD := TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5.ofGaugeGeneratorData D
 617  exact ⟨⟨FD⟩,
 618    ⟨TensorShearSector.PeriodicTTProjectorData5.ofFiniteGeneratorData FD⟩,
 619    TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_gaugeGeneratorData D⟩
 620
 621/-- Track 1.D endpoint: if the gauge map is defined directly from finite
 622generators, then the separate gauge-span proof is automatic. -/
 623def Track1DTTGeneratorMapProjectorReductionEndpoint : Prop :=
 624  ∀ (GIdx : Type) (gFintype : Fintype GIdx),
 625    ∀ D : @TensorShearSector.PeriodicTTGeneratorMapProjectorData5 GIdx gFintype,
 626      Nonempty
 627        (@TensorShearSector.PeriodicTTGaugeGeneratorProjectorData5
 628          (GIdx → ℝ)
 629          (TensorShearSector.periodicGaugeGeneratorMap5 D.gaugeGen)
 630          GIdx gFintype) ∧
 631      Nonempty
 632        (@TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5
 633          (GIdx → ℝ)
 634          (TensorShearSector.periodicGaugeGeneratorMap5 D.gaugeGen)
 635          (Fin TensorShearSector.PeriodicTorus5.K.nV) GIdx
 636          inferInstance gFintype) ∧
 637      Nonempty
 638        (TensorShearSector.PeriodicTTProjectorData5
 639          (GIdx → ℝ) (TensorShearSector.periodicGaugeGeneratorMap5 D.gaugeGen)) ∧
 640      TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
 641        (GIdx → ℝ) (TensorShearSector.periodicGaugeGeneratorMap5 D.gaugeGen)
 642
 643/-- Generator-map projector-data reduction endpoint consumed by Track 7. -/
 644theorem track1D_tt_generator_map_projector_reduction_endpoint_holds :
 645    Track1DTTGeneratorMapProjectorReductionEndpoint := by
 646  intro GIdx gFintype D
 647  letI := gFintype
 648  let GD := TensorShearSector.PeriodicTTGaugeGeneratorProjectorData5.ofGeneratorMapData D
 649  let FD := TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5.ofGaugeGeneratorData GD
 650  exact ⟨⟨GD⟩, ⟨FD⟩,
 651    ⟨TensorShearSector.PeriodicTTProjectorData5.ofFiniteGeneratorData FD⟩,
 652    TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_generatorMapData D⟩
 653
 654/-- Track 1.D endpoint: the concrete vertex-vector longitudinal gauge basis is
 655now the decomposition surface.  What remains is the coefficient projector and
 656reconstruction/orthogonality proof for that basis. -/
 657def Track1DTTLongitudinalProjectorReductionEndpoint : Prop :=
 658  TensorShearSector.PeriodicTTLongitudinalProjectorData5 →
 659    Nonempty
 660      (TensorShearSector.PeriodicTTGeneratorMapProjectorData5
 661        TensorShearSector.PeriodicLongitudinalGaugeIdx5) ∧
 662    Nonempty
 663      (TensorShearSector.PeriodicTTProjectorData5
 664        (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 665        TensorShearSector.periodicLongitudinalGaugeMap5) ∧
 666    TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
 667      (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 668      TensorShearSector.periodicLongitudinalGaugeMap5
 669
 670/-- Concrete longitudinal projector-data reduction endpoint consumed by Track 7. -/
 671theorem track1D_tt_longitudinal_projector_reduction_endpoint_holds :
 672    Track1DTTLongitudinalProjectorReductionEndpoint := by
 673  intro D
 674  let GM := TensorShearSector.PeriodicTTGeneratorMapProjectorData5.ofLongitudinalData D
 675  let GD := TensorShearSector.PeriodicTTGaugeGeneratorProjectorData5.ofGeneratorMapData GM
 676  let FD := TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5.ofGaugeGeneratorData GD
 677  exact ⟨⟨GM⟩,
 678    ⟨TensorShearSector.PeriodicTTProjectorData5.ofFiniteGeneratorData FD⟩,
 679    TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_longitudinalData D⟩
 680
 681/-- Track 1.D endpoint: the remaining concrete TT decomposition input can be
 682given entirely as coefficient projectors on the fixed conformal and longitudinal
 683bases. -/
 684def Track1DTTLongitudinalCoefficientProjectorReductionEndpoint : Prop :=
 685  TensorShearSector.PeriodicTTLongitudinalCoefficientProjectorData5 →
 686    Nonempty TensorShearSector.PeriodicTTLongitudinalProjectorData5 ∧
 687    Nonempty
 688      (TensorShearSector.PeriodicTTGeneratorMapProjectorData5
 689        TensorShearSector.PeriodicLongitudinalGaugeIdx5) ∧
 690    Nonempty
 691      (TensorShearSector.PeriodicTTProjectorData5
 692        (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 693        TensorShearSector.periodicLongitudinalGaugeMap5) ∧
 694    TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
 695      (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 696      TensorShearSector.periodicLongitudinalGaugeMap5
 697
 698/-- Coefficient-projector reduction endpoint consumed by Track 7. -/
 699theorem track1D_tt_longitudinal_coefficient_projector_reduction_endpoint_holds :
 700    Track1DTTLongitudinalCoefficientProjectorReductionEndpoint := by
 701  intro D
 702  let LD := TensorShearSector.PeriodicTTLongitudinalProjectorData5.ofCoefficientData D
 703  let GM := TensorShearSector.PeriodicTTGeneratorMapProjectorData5.ofLongitudinalData LD
 704  let GD := TensorShearSector.PeriodicTTGaugeGeneratorProjectorData5.ofGeneratorMapData GM
 705  let FD := TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5.ofGaugeGeneratorData GD
 706  exact ⟨⟨LD⟩, ⟨GM⟩,
 707    ⟨TensorShearSector.PeriodicTTProjectorData5.ofFiniteGeneratorData FD⟩,
 708    TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_longitudinalCoefficientData D⟩
 709
 710/-- Track 1.D endpoint: the current finite solve only needs conformal and
 711longitudinal coefficient projectors whose residual is orthogonal to both fixed
 712generator families. -/
 713def Track1DTTLongitudinalCoefficientSolutionReductionEndpoint : Prop :=
 714  TensorShearSector.PeriodicTTLongitudinalCoefficientSolutionData5 →
 715    Nonempty TensorShearSector.PeriodicTTLongitudinalCoefficientProjectorData5 ∧
 716    Nonempty TensorShearSector.PeriodicTTLongitudinalProjectorData5 ∧
 717    Nonempty
 718      (TensorShearSector.PeriodicTTProjectorData5
 719        (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 720        TensorShearSector.periodicLongitudinalGaugeMap5) ∧
 721    TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
 722      (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 723      TensorShearSector.periodicLongitudinalGaugeMap5
 724
 725/-- Residual-defined coefficient-solution reduction endpoint consumed by Track 7. -/
 726theorem track1D_tt_longitudinal_coefficient_solution_reduction_endpoint_holds :
 727    Track1DTTLongitudinalCoefficientSolutionReductionEndpoint := by
 728  intro D
 729  let CD := TensorShearSector.PeriodicTTLongitudinalCoefficientProjectorData5.ofSolutionData D
 730  let LD := TensorShearSector.PeriodicTTLongitudinalProjectorData5.ofCoefficientData CD
 731  let GM := TensorShearSector.PeriodicTTGeneratorMapProjectorData5.ofLongitudinalData LD
 732  let GD := TensorShearSector.PeriodicTTGaugeGeneratorProjectorData5.ofGeneratorMapData GM
 733  let FD := TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5.ofGaugeGeneratorData GD
 734  exact ⟨⟨CD⟩, ⟨LD⟩,
 735    ⟨TensorShearSector.PeriodicTTProjectorData5.ofFiniteGeneratorData FD⟩,
 736    TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_longitudinalCoefficientSolutionData D⟩
 737
 738/-- Track 1.D endpoint: a single combined normal-equation solution on the fixed
 739conformal plus longitudinal generator family closes the finite TT decomposition. -/
 740def Track1DTTNormalEquationReductionEndpoint : Prop :=
 741  TensorShearSector.PeriodicTTNormalEquationSolutionData5 →
 742    Nonempty TensorShearSector.PeriodicTTLongitudinalCoefficientSolutionData5 ∧
 743    Nonempty TensorShearSector.PeriodicTTLongitudinalCoefficientProjectorData5 ∧
 744    Nonempty
 745      (TensorShearSector.PeriodicTTProjectorData5
 746        (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 747        TensorShearSector.periodicLongitudinalGaugeMap5) ∧
 748    TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
 749      (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 750      TensorShearSector.periodicLongitudinalGaugeMap5
 751
 752/-- Combined normal-equation reduction endpoint consumed by Track 7. -/
 753theorem track1D_tt_normal_equation_reduction_endpoint_holds :
 754    Track1DTTNormalEquationReductionEndpoint := by
 755  intro D
 756  let SD := TensorShearSector.PeriodicTTLongitudinalCoefficientSolutionData5.ofNormalEquationData D
 757  let CD := TensorShearSector.PeriodicTTLongitudinalCoefficientProjectorData5.ofSolutionData SD
 758  let LD := TensorShearSector.PeriodicTTLongitudinalProjectorData5.ofCoefficientData CD
 759  let GM := TensorShearSector.PeriodicTTGeneratorMapProjectorData5.ofLongitudinalData LD
 760  let GD := TensorShearSector.PeriodicTTGaugeGeneratorProjectorData5.ofGeneratorMapData GM
 761  let FD := TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5.ofGaugeGeneratorData GD
 762  exact ⟨⟨SD⟩, ⟨CD⟩,
 763    ⟨TensorShearSector.PeriodicTTProjectorData5.ofFiniteGeneratorData FD⟩,
 764    TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_normalEquationData D⟩
 765
 766/-- Track 1.D endpoint: solving the explicit finite Gram system supplies the
 767combined normal equations and closes the finite TT decomposition. -/
 768def Track1DTTGramSystemReductionEndpoint : Prop :=
 769  TensorShearSector.PeriodicTTGramSystemSolutionData5 →
 770    Nonempty TensorShearSector.PeriodicTTNormalEquationSolutionData5 ∧
 771    Nonempty TensorShearSector.PeriodicTTLongitudinalCoefficientSolutionData5 ∧
 772    Nonempty
 773      (TensorShearSector.PeriodicTTProjectorData5
 774        (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 775        TensorShearSector.periodicLongitudinalGaugeMap5) ∧
 776    TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
 777      (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 778      TensorShearSector.periodicLongitudinalGaugeMap5
 779
 780/-- Explicit Gram-system reduction endpoint consumed by Track 7. -/
 781theorem track1D_tt_gram_system_reduction_endpoint_holds :
 782    Track1DTTGramSystemReductionEndpoint := by
 783  intro D
 784  let ND := TensorShearSector.PeriodicTTNormalEquationSolutionData5.ofGramSystemData D
 785  let SD := TensorShearSector.PeriodicTTLongitudinalCoefficientSolutionData5.ofNormalEquationData ND
 786  let CD := TensorShearSector.PeriodicTTLongitudinalCoefficientProjectorData5.ofSolutionData SD
 787  let LD := TensorShearSector.PeriodicTTLongitudinalProjectorData5.ofCoefficientData CD
 788  let GM := TensorShearSector.PeriodicTTGeneratorMapProjectorData5.ofLongitudinalData LD
 789  let GD := TensorShearSector.PeriodicTTGaugeGeneratorProjectorData5.ofGeneratorMapData GM
 790  let FD := TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5.ofGaugeGeneratorData GD
 791  exact ⟨⟨ND⟩, ⟨SD⟩,
 792    ⟨TensorShearSector.PeriodicTTProjectorData5.ofFiniteGeneratorData FD⟩,
 793    TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_gramSystemData D⟩
 794
 795/-- Track 1.D endpoint: a finite solver for every load induced by an edge
 796perturbation supplies the Gram system and closes the TT decomposition. -/
 797def Track1DTTGramLoadSolverReductionEndpoint : Prop :=
 798  TensorShearSector.PeriodicTTGramLoadSolverData5 →
 799    Nonempty TensorShearSector.PeriodicTTGramSystemSolutionData5 ∧
 800    Nonempty TensorShearSector.PeriodicTTNormalEquationSolutionData5 ∧
 801    Nonempty
 802      (TensorShearSector.PeriodicTTProjectorData5
 803        (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 804        TensorShearSector.periodicLongitudinalGaugeMap5) ∧
 805    TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
 806      (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 807      TensorShearSector.periodicLongitudinalGaugeMap5
 808
 809/-- Explicit Gram-load solver endpoint consumed by Track 7. -/
 810theorem track1D_tt_gram_load_solver_reduction_endpoint_holds :
 811    Track1DTTGramLoadSolverReductionEndpoint := by
 812  intro D
 813  let GS := TensorShearSector.PeriodicTTGramSystemSolutionData5.ofLoadSolverData D
 814  let ND := TensorShearSector.PeriodicTTNormalEquationSolutionData5.ofGramSystemData GS
 815  let SD := TensorShearSector.PeriodicTTLongitudinalCoefficientSolutionData5.ofNormalEquationData ND
 816  let CD := TensorShearSector.PeriodicTTLongitudinalCoefficientProjectorData5.ofSolutionData SD
 817  let LD := TensorShearSector.PeriodicTTLongitudinalProjectorData5.ofCoefficientData CD
 818  let GM := TensorShearSector.PeriodicTTGeneratorMapProjectorData5.ofLongitudinalData LD
 819  let GD := TensorShearSector.PeriodicTTGaugeGeneratorProjectorData5.ofGeneratorMapData GM
 820  let FD := TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5.ofGaugeGeneratorData GD
 821  exact ⟨⟨GS⟩, ⟨ND⟩,
 822    ⟨TensorShearSector.PeriodicTTProjectorData5.ofFiniteGeneratorData FD⟩,
 823    TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_gramLoadSolverData D⟩
 824
 825/-- Track 1.D endpoint: every physical TT load lies in the image of the finite
 826Gram operator, so a solver on the load subspace exists and closes the TT split. -/
 827def Track1DTTGramLoadImageReductionEndpoint : Prop :=
 828  TensorShearSector.PeriodicTTGramLoadImageData5 →
 829    Nonempty TensorShearSector.PeriodicTTGramLoadSolverData5 ∧
 830    Nonempty TensorShearSector.PeriodicTTGramSystemSolutionData5 ∧
 831    Nonempty
 832      (TensorShearSector.PeriodicTTProjectorData5
 833        (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 834        TensorShearSector.periodicLongitudinalGaugeMap5) ∧
 835    TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
 836      (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 837      TensorShearSector.periodicLongitudinalGaugeMap5
 838
 839/-- Explicit Gram-load image endpoint consumed by Track 7. -/
 840theorem track1D_tt_gram_load_image_reduction_endpoint_holds :
 841    Track1DTTGramLoadImageReductionEndpoint := by
 842  intro D
 843  let LS := TensorShearSector.PeriodicTTGramLoadSolverData5.ofLoadImageData D
 844  let GS := TensorShearSector.PeriodicTTGramSystemSolutionData5.ofLoadSolverData LS
 845  let ND := TensorShearSector.PeriodicTTNormalEquationSolutionData5.ofGramSystemData GS
 846  let SD := TensorShearSector.PeriodicTTLongitudinalCoefficientSolutionData5.ofNormalEquationData ND
 847  let CD := TensorShearSector.PeriodicTTLongitudinalCoefficientProjectorData5.ofSolutionData SD
 848  let LD := TensorShearSector.PeriodicTTLongitudinalProjectorData5.ofCoefficientData CD
 849  let GM := TensorShearSector.PeriodicTTGeneratorMapProjectorData5.ofLongitudinalData LD
 850  let GD := TensorShearSector.PeriodicTTGaugeGeneratorProjectorData5.ofGeneratorMapData GM
 851  let FD := TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5.ofGaugeGeneratorData GD
 852  exact ⟨⟨LS⟩, ⟨GS⟩,
 853    ⟨TensorShearSector.PeriodicTTProjectorData5.ofFiniteGeneratorData FD⟩,
 854    TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_gramLoadImageData D⟩
 855
 856/-- Track 1.D endpoint: the finite Gram-kernel criterion implies every physical
 857TT load lies in the Gram image and closes the TT split. -/
 858def Track1DTTGramKernelCriterionReductionEndpoint : Prop :=
 859  TensorShearSector.PeriodicTTGramKernelCriterionData5 →
 860    Nonempty TensorShearSector.PeriodicTTGramLoadImageData5 ∧
 861    Nonempty TensorShearSector.PeriodicTTGramLoadSolverData5 ∧
 862    Nonempty
 863      (TensorShearSector.PeriodicTTProjectorData5
 864        (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 865        TensorShearSector.periodicLongitudinalGaugeMap5) ∧
 866    TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
 867      (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 868      TensorShearSector.periodicLongitudinalGaugeMap5
 869
 870/-- Explicit Gram-kernel criterion endpoint consumed by Track 7. -/
 871theorem track1D_tt_gram_kernel_criterion_reduction_endpoint_holds :
 872    Track1DTTGramKernelCriterionReductionEndpoint := by
 873  intro D
 874  let LI := TensorShearSector.PeriodicTTGramLoadImageData5.ofKernelCriterionData D
 875  let LS := TensorShearSector.PeriodicTTGramLoadSolverData5.ofLoadImageData LI
 876  let GS := TensorShearSector.PeriodicTTGramSystemSolutionData5.ofLoadSolverData LS
 877  let ND := TensorShearSector.PeriodicTTNormalEquationSolutionData5.ofGramSystemData GS
 878  let SD := TensorShearSector.PeriodicTTLongitudinalCoefficientSolutionData5.ofNormalEquationData ND
 879  let CD := TensorShearSector.PeriodicTTLongitudinalCoefficientProjectorData5.ofSolutionData SD
 880  let LD := TensorShearSector.PeriodicTTLongitudinalProjectorData5.ofCoefficientData CD
 881  let GM := TensorShearSector.PeriodicTTGeneratorMapProjectorData5.ofLongitudinalData LD
 882  let GD := TensorShearSector.PeriodicTTGaugeGeneratorProjectorData5.ofGeneratorMapData GM
 883  let FD := TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5.ofGaugeGeneratorData GD
 884  exact ⟨⟨LI⟩, ⟨LS⟩,
 885    ⟨TensorShearSector.PeriodicTTProjectorData5.ofFiniteGeneratorData FD⟩,
 886    TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_gramKernelCriterionData D⟩
 887
 888/-- Track 1.D endpoint: if Gram-kernel coefficient vectors generate the zero
 889edge perturbation, the load-annihilates-kernel half of the finite criterion is
 890automatic. -/
 891def Track1DTTGramKernelGeneratorMapZeroReductionEndpoint : Prop :=
 892  TensorShearSector.PeriodicTTGramKernelGeneratorMapZeroData5 →
 893    Nonempty TensorShearSector.PeriodicTTGramKernelCriterionData5 ∧
 894    Nonempty TensorShearSector.PeriodicTTGramLoadImageData5 ∧
 895    Nonempty
 896      (TensorShearSector.PeriodicTTProjectorData5
 897        (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 898        TensorShearSector.periodicLongitudinalGaugeMap5) ∧
 899    TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
 900      (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 901      TensorShearSector.periodicLongitudinalGaugeMap5
 902
 903/-- Explicit Gram-kernel generator-map-zero endpoint consumed by Track 7. -/
 904theorem track1D_tt_gram_kernel_generator_map_zero_reduction_endpoint_holds :
 905    Track1DTTGramKernelGeneratorMapZeroReductionEndpoint := by
 906  intro D
 907  let KC := TensorShearSector.PeriodicTTGramKernelCriterionData5.ofKernelGeneratorMapZeroData D
 908  let LI := TensorShearSector.PeriodicTTGramLoadImageData5.ofKernelCriterionData KC
 909  let LS := TensorShearSector.PeriodicTTGramLoadSolverData5.ofLoadImageData LI
 910  let GS := TensorShearSector.PeriodicTTGramSystemSolutionData5.ofLoadSolverData LS
 911  let ND := TensorShearSector.PeriodicTTNormalEquationSolutionData5.ofGramSystemData GS
 912  let SD := TensorShearSector.PeriodicTTLongitudinalCoefficientSolutionData5.ofNormalEquationData ND
 913  let CD := TensorShearSector.PeriodicTTLongitudinalCoefficientProjectorData5.ofSolutionData SD
 914  let LD := TensorShearSector.PeriodicTTLongitudinalProjectorData5.ofCoefficientData CD
 915  let GM := TensorShearSector.PeriodicTTGeneratorMapProjectorData5.ofLongitudinalData LD
 916  let GD := TensorShearSector.PeriodicTTGaugeGeneratorProjectorData5.ofGeneratorMapData GM
 917  let FD := TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5.ofGaugeGeneratorData GD
 918  exact ⟨⟨KC⟩, ⟨LI⟩,
 919    ⟨TensorShearSector.PeriodicTTProjectorData5.ofFiniteGeneratorData FD⟩,
 920    TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_gramKernelGeneratorMapZeroData D⟩
 921
 922/-- Track 1.D endpoint: the finite Gram range/Fredholm criterion alone closes the
 923TT split, since Gram-kernel coefficients are proved to generate zero. -/
 924def Track1DTTGramRangeCriterionReductionEndpoint : Prop :=
 925  TensorShearSector.PeriodicTTGramRangeCriterionData5 →
 926    Nonempty TensorShearSector.PeriodicTTGramKernelCriterionData5 ∧
 927    Nonempty TensorShearSector.PeriodicTTGramLoadImageData5 ∧
 928    Nonempty
 929      (TensorShearSector.PeriodicTTProjectorData5
 930        (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 931        TensorShearSector.periodicLongitudinalGaugeMap5) ∧
 932    TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
 933      (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 934      TensorShearSector.periodicLongitudinalGaugeMap5
 935
 936/-- Explicit Gram range-criterion endpoint consumed by Track 7. -/
 937theorem track1D_tt_gram_range_criterion_reduction_endpoint_holds :
 938    Track1DTTGramRangeCriterionReductionEndpoint := by
 939  intro D
 940  let KC := TensorShearSector.PeriodicTTGramKernelCriterionData5.ofRangeCriterionData D
 941  let LI := TensorShearSector.PeriodicTTGramLoadImageData5.ofKernelCriterionData KC
 942  let LS := TensorShearSector.PeriodicTTGramLoadSolverData5.ofLoadImageData LI
 943  let GS := TensorShearSector.PeriodicTTGramSystemSolutionData5.ofLoadSolverData LS
 944  let ND := TensorShearSector.PeriodicTTNormalEquationSolutionData5.ofGramSystemData GS
 945  let SD := TensorShearSector.PeriodicTTLongitudinalCoefficientSolutionData5.ofNormalEquationData ND
 946  let CD := TensorShearSector.PeriodicTTLongitudinalCoefficientProjectorData5.ofSolutionData SD
 947  let LD := TensorShearSector.PeriodicTTLongitudinalProjectorData5.ofCoefficientData CD
 948  let GM := TensorShearSector.PeriodicTTGeneratorMapProjectorData5.ofLongitudinalData LD
 949  let GD := TensorShearSector.PeriodicTTGaugeGeneratorProjectorData5.ofGeneratorMapData GM
 950  let FD := TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5.ofGaugeGeneratorData GD
 951  exact ⟨⟨KC⟩, ⟨LI⟩,
 952    ⟨TensorShearSector.PeriodicTTProjectorData5.ofFiniteGeneratorData FD⟩,
 953    TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_gramRangeCriterionData D⟩
 954
 955/-- Track 1.D endpoint: the fixed finite TT Gram operator is self-adjoint for
 956the coefficient-space inner product. -/
 957def Track1DTTGramSelfAdjointEndpoint : Prop :=
 958  ∀ a b : TensorShearSector.PeriodicTTNormalEquationIdx5 → ℝ,
 959    TensorShearSector.periodicTTNormalEquationCoeffInnerProduct5
 960      (TensorShearSector.periodicTTNormalEquationGramVector5 a) b =
 961    TensorShearSector.periodicTTNormalEquationCoeffInnerProduct5
 962      a (TensorShearSector.periodicTTNormalEquationGramVector5 b)
 963
 964/-- Explicit Gram self-adjointness endpoint consumed by Track 7. -/
 965theorem track1D_tt_gram_self_adjoint_endpoint_holds :
 966    Track1DTTGramSelfAdjointEndpoint :=
 967  TensorShearSector.periodicTTNormalEquationGram_selfAdjoint5
 968
 969/-- Track 1.D endpoint: the finite TT Gram range criterion is proved for the
 970fixed combined conformal plus longitudinal generator family, so the concrete
 971TT projector split closes at `N=5`. -/
 972def Track1DTTGramRangeClosedEndpoint : Prop :=
 973  Nonempty TensorShearSector.PeriodicTTGramRangeCriterionData5 ∧
 974  Nonempty TensorShearSector.PeriodicTTGramKernelCriterionData5 ∧
 975  Nonempty TensorShearSector.PeriodicTTGramLoadImageData5 ∧
 976  Nonempty
 977    (TensorShearSector.PeriodicTTProjectorData5
 978      (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 979      TensorShearSector.periodicLongitudinalGaugeMap5) ∧
 980  TensorShearSector.PeriodicFreudenthalTTOrthogonalDecompositionTargetAtN5
 981    (TensorShearSector.PeriodicLongitudinalGaugeIdx5 → ℝ)
 982    TensorShearSector.periodicLongitudinalGaugeMap5
 983
 984/-- Proved finite Gram range endpoint consumed by Track 7. -/
 985theorem track1D_tt_gram_range_closed_endpoint_holds :
 986    Track1DTTGramRangeClosedEndpoint := by
 987  let D := TensorShearSector.periodicTTGramRangeCriterionData5_proved
 988  let KC := TensorShearSector.PeriodicTTGramKernelCriterionData5.ofRangeCriterionData D
 989  let LI := TensorShearSector.PeriodicTTGramLoadImageData5.ofKernelCriterionData KC
 990  let LS := TensorShearSector.PeriodicTTGramLoadSolverData5.ofLoadImageData LI
 991  let GS := TensorShearSector.PeriodicTTGramSystemSolutionData5.ofLoadSolverData LS
 992  let ND := TensorShearSector.PeriodicTTNormalEquationSolutionData5.ofGramSystemData GS
 993  let SD := TensorShearSector.PeriodicTTLongitudinalCoefficientSolutionData5.ofNormalEquationData ND
 994  let CD := TensorShearSector.PeriodicTTLongitudinalCoefficientProjectorData5.ofSolutionData SD
 995  let LD := TensorShearSector.PeriodicTTLongitudinalProjectorData5.ofCoefficientData CD
 996  let GM := TensorShearSector.PeriodicTTGeneratorMapProjectorData5.ofLongitudinalData LD
 997  let GD := TensorShearSector.PeriodicTTGaugeGeneratorProjectorData5.ofGeneratorMapData GM
 998  let FD := TensorShearSector.PeriodicTTFiniteGeneratorProjectorData5.ofGaugeGeneratorData GD
 999  exact ⟨⟨D⟩, ⟨KC⟩, ⟨LI⟩,
1000    ⟨TensorShearSector.PeriodicTTProjectorData5.ofFiniteGeneratorData FD⟩,
1001    TensorShearSector.periodicFreudenthalTTOrthogonalDecompositionTargetAtN5_of_provedGramRangeCriterion⟩
1002
1003/-- Track 1.D endpoint: once the Regge TT Hessian operator and lattice
1004Lichnerowicz operator are identified pointwise on TT modes, the bilinear and
1005quadratic TT energy matches follow. -/
1006def Track1DTTHessianLichnerowiczBilinearReductionEndpoint : Prop :=
1007  ∀ (D : TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5)
1008    (ε η : TensorShearSector.PeriodicEdgePerturbation5),
1009    TensorShearSector.PeriodicLongitudinalTTSubspace5 ε →
1010    TensorShearSector.PeriodicLongitudinalTTSubspace5 η →
1011      TensorShearSector.periodicTTOperatorBilinear5 D.reggeHessianTT ε η =
1012        TensorShearSector.periodicTTOperatorBilinear5 D.latticeLichnerowiczTT ε η ∧
1013      TensorShearSector.periodicTTOperatorBilinear5 D.reggeHessianTT ε ε =
1014        TensorShearSector.periodicTTOperatorBilinear5 D.latticeLichnerowiczTT ε ε
1015
1016/-- TT Hessian/Lichnerowicz bilinear reduction endpoint consumed by Track 7. -/
1017theorem track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds :
1018    Track1DTTHessianLichnerowiczBilinearReductionEndpoint := by
1019  intro D ε η hε hη
1020  exact ⟨
1021    TensorShearSector.periodicTTHessianLichnerowicz_bilinear_eq_of_matchData5
1022      D ε η hη,
1023    TensorShearSector.periodicTTHessianLichnerowicz_quadratic_eq_of_matchData5
1024      D ε hε⟩
1025
1026/-- Track 1.D endpoint: rowwise equality of the Regge TT Hessian edge kernel and
1027the lattice Lichnerowicz edge kernel on TT perturbations supplies the
1028operator-match data and hence the bilinear/quadratic TT energy matches. -/
1029def Track1DTTHessianLichnerowiczKernelRowReductionEndpoint : Prop :=
1030  TensorShearSector.PeriodicTTHessianLichnerowiczKernelRowData5 →
1031    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5 ∧
1032    Track1DTTHessianLichnerowiczBilinearReductionEndpoint
1033
1034/-- Rowwise edge-kernel reduction endpoint consumed by Track 7. -/
1035theorem track1D_tt_hessian_lichnerowicz_kernel_row_reduction_endpoint_holds :
1036    Track1DTTHessianLichnerowiczKernelRowReductionEndpoint := by
1037  intro D
1038  exact ⟨
1039    ⟨TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5.ofKernelRowData D⟩,
1040    track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds⟩
1041
1042/-- Track 1.D endpoint: entrywise equality of the two finite edge kernels is a
1043stronger stencil-level sufficient condition for the TT Hessian/Lichnerowicz
1044match. -/
1045def Track1DTTHessianLichnerowiczKernelEntryReductionEndpoint : Prop :=
1046  TensorShearSector.PeriodicTTHessianLichnerowiczKernelEntryData5 →
1047    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczKernelRowData5 ∧
1048    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5 ∧
1049    Track1DTTHessianLichnerowiczBilinearReductionEndpoint
1050
1051/-- Entrywise edge-kernel reduction endpoint consumed by Track 7. -/
1052theorem track1D_tt_hessian_lichnerowicz_kernel_entry_reduction_endpoint_holds :
1053    Track1DTTHessianLichnerowiczKernelEntryReductionEndpoint := by
1054  intro D
1055  let R := TensorShearSector.PeriodicTTHessianLichnerowiczKernelRowData5.ofEntryData D
1056  exact ⟨
1057    ⟨R⟩,
1058    ⟨TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5.ofKernelEntryData D⟩,
1059    track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds⟩
1060
1061/-- Track 1.D endpoint: it is enough to prove that the residual edge kernel
1062`ReggeTT - LichnerowiczTT` annihilates every TT perturbation. -/
1063def Track1DTTHessianLichnerowiczResidualTTZeroReductionEndpoint : Prop :=
1064  TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5 →
1065    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczKernelRowData5 ∧
1066    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5 ∧
1067    Track1DTTHessianLichnerowiczBilinearReductionEndpoint
1068
1069/-- Residual-kernel zero endpoint consumed by Track 7. -/
1070theorem track1D_tt_hessian_lichnerowicz_residual_tt_zero_reduction_endpoint_holds :
1071    Track1DTTHessianLichnerowiczResidualTTZeroReductionEndpoint := by
1072  intro D
1073  let R := TensorShearSector.PeriodicTTHessianLichnerowiczKernelRowData5.ofResidualTTZeroData D
1074  exact ⟨
1075    ⟨R⟩,
1076    ⟨TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5.ofResidualTTZeroData D⟩,
1077    track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds⟩
1078
1079/-- Track 1.D endpoint: it is enough to prove every residual edge-kernel row
1080lies in the combined conformal plus longitudinal generator span. TT
1081orthogonality then annihilates the residual. -/
1082def Track1DTTHessianLichnerowiczResidualRowSpanReductionEndpoint : Prop :=
1083  TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowSpanData5 →
1084    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5 ∧
1085    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczKernelRowData5 ∧
1086    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5 ∧
1087    Track1DTTHessianLichnerowiczBilinearReductionEndpoint
1088
1089/-- Residual-row generator-span endpoint consumed by Track 7. -/
1090theorem track1D_tt_hessian_lichnerowicz_residual_row_span_reduction_endpoint_holds :
1091    Track1DTTHessianLichnerowiczResidualRowSpanReductionEndpoint := by
1092  intro D
1093  let Z :=
1094    TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5.ofResidualRowSpanData D
1095  let R := TensorShearSector.PeriodicTTHessianLichnerowiczKernelRowData5.ofResidualTTZeroData Z
1096  exact ⟨
1097    ⟨Z⟩,
1098    ⟨R⟩,
1099    ⟨TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5.ofResidualRowSpanData D⟩,
1100    track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds⟩
1101
1102/-- Track 1.D endpoint: explicit residual-row coefficient data is enough to
1103close the TT Hessian/Lichnerowicz consequence route. -/
1104def Track1DTTHessianLichnerowiczResidualRowCoeffReductionEndpoint : Prop :=
1105  TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowCoeffData5 →
1106    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowSpanData5 ∧
1107    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5 ∧
1108    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczKernelRowData5 ∧
1109    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5 ∧
1110    Track1DTTHessianLichnerowiczBilinearReductionEndpoint
1111
1112/-- Explicit residual-row coefficient endpoint consumed by Track 7. -/
1113theorem track1D_tt_hessian_lichnerowicz_residual_row_coeff_reduction_endpoint_holds :
1114    Track1DTTHessianLichnerowiczResidualRowCoeffReductionEndpoint := by
1115  intro D
1116  let S := TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowSpanData5.ofRowCoeffData D
1117  let Z := TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5.ofRowCoeffData D
1118  let R := TensorShearSector.PeriodicTTHessianLichnerowiczKernelRowData5.ofResidualTTZeroData Z
1119  exact ⟨
1120    ⟨S⟩,
1121    ⟨Z⟩,
1122    ⟨R⟩,
1123    ⟨TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5.ofRowCoeffData D⟩,
1124    track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds⟩
1125
1126/-- Track 1.D endpoint: entrywise residual-row coefficient identities are enough
1127to close the TT Hessian/Lichnerowicz consequence route. -/
1128def Track1DTTHessianLichnerowiczResidualRowCoeffEntryReductionEndpoint : Prop :=
1129  TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowCoeffEntryData5 →
1130    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowCoeffData5 ∧
1131    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowSpanData5 ∧
1132    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5 ∧
1133    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczKernelRowData5 ∧
1134    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5 ∧
1135    Track1DTTHessianLichnerowiczBilinearReductionEndpoint
1136
1137/-- Entrywise residual-row coefficient endpoint consumed by Track 7. -/
1138theorem track1D_tt_hessian_lichnerowicz_residual_row_coeff_entry_reduction_endpoint_holds :
1139    Track1DTTHessianLichnerowiczResidualRowCoeffEntryReductionEndpoint := by
1140  intro D
1141  let C := TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowCoeffData5.ofEntryData D
1142  let S := TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowSpanData5.ofEntryCoeffData D
1143  let Z := TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5.ofEntryCoeffData D
1144  let R := TensorShearSector.PeriodicTTHessianLichnerowiczKernelRowData5.ofResidualTTZeroData Z
1145  exact ⟨
1146    ⟨C⟩,
1147    ⟨S⟩,
1148    ⟨Z⟩,
1149    ⟨R⟩,
1150    ⟨TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5.ofEntryCoeffData D⟩,
1151    track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds⟩
1152
1153/-- Track 1.D endpoint: raw scalar formulas
1154`Regge(e,f) - Lichnerowicz(e,f) = generatorCoeff(e)(f)` are enough to close
1155the TT Hessian/Lichnerowicz consequence route. -/
1156def Track1DTTHessianLichnerowiczResidualEntryFormulaReductionEndpoint : Prop :=
1157  TensorShearSector.PeriodicTTHessianLichnerowiczResidualEntryFormulaData5 →
1158    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowCoeffEntryData5 ∧
1159    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowCoeffData5 ∧
1160    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowSpanData5 ∧
1161    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5 ∧
1162    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczKernelRowData5 ∧
1163    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5 ∧
1164    Track1DTTHessianLichnerowiczBilinearReductionEndpoint
1165
1166/-- Raw scalar residual formula endpoint consumed by Track 7. -/
1167theorem track1D_tt_hessian_lichnerowicz_residual_entry_formula_reduction_endpoint_holds :
1168    Track1DTTHessianLichnerowiczResidualEntryFormulaReductionEndpoint := by
1169  intro D
1170  let E := TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowCoeffEntryData5.ofFormulaData D
1171  let C := TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowCoeffData5.ofFormulaData D
1172  let S := TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowSpanData5.ofFormulaData D
1173  let Z := TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5.ofFormulaData D
1174  let R := TensorShearSector.PeriodicTTHessianLichnerowiczKernelRowData5.ofResidualTTZeroData Z
1175  exact ⟨
1176    ⟨E⟩,
1177    ⟨C⟩,
1178    ⟨S⟩,
1179    ⟨Z⟩,
1180    ⟨R⟩,
1181    ⟨TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5.ofFormulaData D⟩,
1182    track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds⟩
1183
1184/-- Track 1.D endpoint: encoded `Fin K.nE` scalar formulas feed the typed
1185periodic-edge residual formula route through `PeriodicTorus5.edgeEquiv`. -/
1186def Track1DTTHessianLichnerowiczEncodedResidualEntryFormulaReductionEndpoint : Prop :=
1187  TensorShearSector.EncodedTTHessianLichnerowiczResidualEntryFormulaData5 →
1188    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualEntryFormulaData5 ∧
1189    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowCoeffEntryData5 ∧
1190    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowCoeffData5 ∧
1191    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowSpanData5 ∧
1192    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5 ∧
1193    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczKernelRowData5 ∧
1194    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5 ∧
1195    Track1DTTHessianLichnerowiczBilinearReductionEndpoint
1196
1197/-- Encoded raw scalar residual formula endpoint consumed by Track 7. -/
1198theorem track1D_tt_hessian_lichnerowicz_encoded_residual_entry_formula_reduction_endpoint_holds :
1199    Track1DTTHessianLichnerowiczEncodedResidualEntryFormulaReductionEndpoint := by
1200  intro D
1201  let F := TensorShearSector.PeriodicTTHessianLichnerowiczResidualEntryFormulaData5.ofEncodedData D
1202  let E := TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowCoeffEntryData5.ofEncodedData D
1203  let C := TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowCoeffData5.ofEncodedData D
1204  let S := TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowSpanData5.ofEncodedData D
1205  let Z := TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5.ofEncodedData D
1206  let R := TensorShearSector.PeriodicTTHessianLichnerowiczKernelRowData5.ofResidualTTZeroData Z
1207  exact ⟨
1208    ⟨F⟩,
1209    ⟨E⟩,
1210    ⟨C⟩,
1211    ⟨S⟩,
1212    ⟨Z⟩,
1213    ⟨R⟩,
1214    ⟨TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5.ofEncodedData D⟩,
1215    track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds⟩
1216
1217/-- Track 1.D endpoint: an encoded residual-kernel certificate feeds the encoded
1218raw scalar formula route. -/
1219def Track1DTTHessianLichnerowiczEncodedResidualKernelFormulaReductionEndpoint : Prop :=
1220  TensorShearSector.EncodedTTHessianLichnerowiczResidualKernelFormulaData5 →
1221    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualEntryFormulaData5 ∧
1222    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualEntryFormulaData5 ∧
1223    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowCoeffEntryData5 ∧
1224    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5 ∧
1225    Track1DTTHessianLichnerowiczBilinearReductionEndpoint
1226
1227/-- Encoded residual-kernel formula endpoint consumed by Track 7. -/
1228theorem track1D_tt_hessian_lichnerowicz_encoded_residual_kernel_formula_reduction_endpoint_holds :
1229    Track1DTTHessianLichnerowiczEncodedResidualKernelFormulaReductionEndpoint := by
1230  intro D
1231  let E := TensorShearSector.EncodedTTHessianLichnerowiczResidualEntryFormulaData5.ofResidualKernelData D
1232  let F := TensorShearSector.PeriodicTTHessianLichnerowiczResidualEntryFormulaData5.ofEncodedResidualKernelData D
1233  let C := TensorShearSector.PeriodicTTHessianLichnerowiczResidualRowCoeffEntryData5.ofEncodedResidualKernelData D
1234  exact ⟨
1235    ⟨E⟩,
1236    ⟨F⟩,
1237    ⟨C⟩,
1238    ⟨TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5.ofEncodedResidualKernelData D⟩,
1239    track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds⟩
1240
1241/-- Track 1.D endpoint: a displacement-row residual-kernel certificate feeds the
1242encoded residual-kernel route. -/
1243def Track1DTTHessianLichnerowiczEncodedResidualDispRowFormulaReductionEndpoint : Prop :=
1244  TensorShearSector.EncodedTTHessianLichnerowiczResidualDispRowFormulaData5 →
1245    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualKernelFormulaData5 ∧
1246    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualEntryFormulaData5 ∧
1247    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualEntryFormulaData5 ∧
1248    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5 ∧
1249    Track1DTTHessianLichnerowiczBilinearReductionEndpoint
1250
1251/-- Displacement-row residual formula endpoint consumed by Track 7. -/
1252theorem track1D_tt_hessian_lichnerowicz_encoded_residual_disp_row_formula_reduction_endpoint_holds :
1253    Track1DTTHessianLichnerowiczEncodedResidualDispRowFormulaReductionEndpoint := by
1254  intro D
1255  let K := TensorShearSector.EncodedTTHessianLichnerowiczResidualKernelFormulaData5.ofDispRowData D
1256  let E := TensorShearSector.EncodedTTHessianLichnerowiczResidualEntryFormulaData5.ofDispRowData D
1257  let F := TensorShearSector.PeriodicTTHessianLichnerowiczResidualEntryFormulaData5.ofDispRowData D
1258  exact ⟨
1259    ⟨K⟩,
1260    ⟨E⟩,
1261    ⟨F⟩,
1262    ⟨TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5.ofDispRowData D⟩,
1263    track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds⟩
1264
1265/-- Track 1.D endpoint: a seven-row origin-table certificate feeds the
1266displacement-row residual route. -/
1267def Track1DTTHessianLichnerowiczEncodedResidualOriginRowTableReductionEndpoint : Prop :=
1268  TensorShearSector.EncodedTTHessianLichnerowiczResidualOriginRowTableFormulaData5 →
1269    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualDispRowFormulaData5 ∧
1270    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualKernelFormulaData5 ∧
1271    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualEntryFormulaData5 ∧
1272    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualEntryFormulaData5 ∧
1273    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5 ∧
1274    Track1DTTHessianLichnerowiczBilinearReductionEndpoint
1275
1276/-- Seven-row origin-table formula endpoint consumed by Track 7. -/
1277theorem track1D_tt_hessian_lichnerowicz_encoded_residual_origin_row_table_reduction_endpoint_holds :
1278    Track1DTTHessianLichnerowiczEncodedResidualOriginRowTableReductionEndpoint := by
1279  intro D
1280  let T := TensorShearSector.EncodedTTHessianLichnerowiczResidualDispRowFormulaData5.ofOriginRowTableData D
1281  let K := TensorShearSector.EncodedTTHessianLichnerowiczResidualKernelFormulaData5.ofOriginRowTableData D
1282  let E := TensorShearSector.EncodedTTHessianLichnerowiczResidualEntryFormulaData5.ofOriginRowTableData D
1283  let F := TensorShearSector.PeriodicTTHessianLichnerowiczResidualEntryFormulaData5.ofOriginRowTableData D
1284  exact ⟨
1285    ⟨T⟩,
1286    ⟨K⟩,
1287    ⟨E⟩,
1288    ⟨F⟩,
1289    ⟨TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5.ofOriginRowTableData D⟩,
1290    track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds⟩
1291
1292/-- Track 1.D endpoint: a typed-column origin-table certificate feeds the
1293seven-row origin-table residual route. -/
1294def Track1DTTHessianLichnerowiczEncodedResidualOriginColumnTableReductionEndpoint : Prop :=
1295  TensorShearSector.EncodedTTHessianLichnerowiczResidualOriginColumnTableFormulaData5 →
1296    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualOriginRowTableFormulaData5 ∧
1297    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualKernelFormulaData5 ∧
1298    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualEntryFormulaData5 ∧
1299    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualEntryFormulaData5 ∧
1300    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5 ∧
1301    Track1DTTHessianLichnerowiczBilinearReductionEndpoint
1302
1303/-- Typed-column origin-table formula endpoint consumed by Track 7. -/
1304theorem track1D_tt_hessian_lichnerowicz_encoded_residual_origin_column_table_reduction_endpoint_holds :
1305    Track1DTTHessianLichnerowiczEncodedResidualOriginColumnTableReductionEndpoint := by
1306  intro D
1307  let T := TensorShearSector.EncodedTTHessianLichnerowiczResidualOriginRowTableFormulaData5.ofOriginColumnTableData D
1308  let K := TensorShearSector.EncodedTTHessianLichnerowiczResidualKernelFormulaData5.ofOriginColumnTableData D
1309  let E := TensorShearSector.EncodedTTHessianLichnerowiczResidualEntryFormulaData5.ofOriginColumnTableData D
1310  let F := TensorShearSector.PeriodicTTHessianLichnerowiczResidualEntryFormulaData5.ofOriginColumnTableData D
1311  exact ⟨
1312    ⟨T⟩,
1313    ⟨K⟩,
1314    ⟨E⟩,
1315    ⟨F⟩,
1316    ⟨TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5.ofOriginColumnTableData D⟩,
1317    track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds⟩
1318
1319/-- Track 1.D endpoint: a raw typed-column residual certificate feeds the
1320typed-column origin-table residual route without requiring an explicit residual
1321matrix input. -/
1322def Track1DTTHessianLichnerowiczEncodedRawOriginColumnReductionEndpoint : Prop :=
1323  TensorShearSector.EncodedTTHessianLichnerowiczRawOriginColumnFormulaData5 →
1324    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualOriginColumnTableFormulaData5 ∧
1325    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualOriginRowTableFormulaData5 ∧
1326    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualKernelFormulaData5 ∧
1327    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualEntryFormulaData5 ∧
1328    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5 ∧
1329    Track1DTTHessianLichnerowiczBilinearReductionEndpoint
1330
1331/-- Raw typed-column residual endpoint consumed by Track 7. -/
1332theorem track1D_tt_hessian_lichnerowicz_encoded_raw_origin_column_reduction_endpoint_holds :
1333    Track1DTTHessianLichnerowiczEncodedRawOriginColumnReductionEndpoint := by
1334  intro D
1335  let C := TensorShearSector.EncodedTTHessianLichnerowiczResidualOriginColumnTableFormulaData5.ofRawOriginColumnData D
1336  let T := TensorShearSector.EncodedTTHessianLichnerowiczResidualOriginRowTableFormulaData5.ofRawOriginColumnData D
1337  let K := TensorShearSector.EncodedTTHessianLichnerowiczResidualKernelFormulaData5.ofRawOriginColumnData D
1338  let F := TensorShearSector.PeriodicTTHessianLichnerowiczResidualEntryFormulaData5.ofRawOriginColumnData D
1339  exact ⟨
1340    ⟨C⟩,
1341    ⟨T⟩,
1342    ⟨K⟩,
1343    ⟨F⟩,
1344    ⟨TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5.ofRawOriginColumnData D⟩,
1345    track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds⟩
1346
1347/-- Track 1.D endpoint: a coefficient-only origin-column certificate feeds the
1348raw typed-column residual route without storing the residual origin table. -/
1349def Track1DTTHessianLichnerowiczEncodedCoeffOriginColumnReductionEndpoint : Prop :=
1350  TensorShearSector.EncodedTTHessianLichnerowiczCoeffOriginColumnFormulaData5 →
1351    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczRawOriginColumnFormulaData5 ∧
1352    Track1DTTHessianLichnerowiczEncodedRawOriginColumnReductionEndpoint
1353
1354/-- Coefficient-only origin-column residual endpoint consumed by Track 7. -/
1355theorem track1D_tt_hessian_lichnerowicz_encoded_coeff_origin_column_reduction_endpoint_holds :
1356    Track1DTTHessianLichnerowiczEncodedCoeffOriginColumnReductionEndpoint := by
1357  intro D
1358  exact ⟨
1359    ⟨TensorShearSector.EncodedTTHessianLichnerowiczRawOriginColumnFormulaData5.ofCoeffOriginColumnData D⟩,
1360    track1D_tt_hessian_lichnerowicz_encoded_raw_origin_column_reduction_endpoint_holds⟩
1361
1362/-- Track 1.D endpoint: a translated coefficient-only certificate feeds the
1363coefficient-origin route by deriving the origin-column scalar formulas from the
1364translated residual formula. -/
1365def Track1DTTHessianLichnerowiczEncodedCoeffTranslatedReductionEndpoint : Prop :=
1366  TensorShearSector.EncodedTTHessianLichnerowiczCoeffTranslatedFormulaData5 →
1367    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczCoeffOriginColumnFormulaData5 ∧
1368    Track1DTTHessianLichnerowiczEncodedCoeffOriginColumnReductionEndpoint
1369
1370/-- Translated coefficient-only residual endpoint consumed by Track 7. -/
1371theorem track1D_tt_hessian_lichnerowicz_encoded_coeff_translated_reduction_endpoint_holds :
1372    Track1DTTHessianLichnerowiczEncodedCoeffTranslatedReductionEndpoint := by
1373  intro D
1374  exact ⟨
1375    ⟨TensorShearSector.EncodedTTHessianLichnerowiczCoeffOriginColumnFormulaData5.ofCoeffTranslatedData D⟩,
1376    track1D_tt_hessian_lichnerowicz_encoded_coeff_origin_column_reduction_endpoint_holds⟩
1377
1378/-- Track 1.D endpoint: a translated coefficient-only certificate exposes the
1379whole finite TT Hessian/Lichnerowicz reduction chain in one audit target. -/
1380def Track1DTTHessianLichnerowiczEncodedCoeffTranslatedFullChainEndpoint : Prop :=
1381  TensorShearSector.EncodedTTHessianLichnerowiczCoeffTranslatedFormulaData5 →
1382    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczCoeffOriginColumnFormulaData5 ∧
1383    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczRawOriginColumnFormulaData5 ∧
1384    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualOriginColumnTableFormulaData5 ∧
1385    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualOriginRowTableFormulaData5 ∧
1386    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualKernelFormulaData5 ∧
1387    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczResidualEntryFormulaData5 ∧
1388    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczResidualEntryFormulaData5 ∧
1389    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5 ∧
1390    Track1DTTHessianLichnerowiczBilinearReductionEndpoint
1391
1392/-- Full-chain translated coefficient-only residual endpoint consumed by Track 7. -/
1393theorem track1D_tt_hessian_lichnerowicz_encoded_coeff_translated_full_chain_endpoint_holds :
1394    Track1DTTHessianLichnerowiczEncodedCoeffTranslatedFullChainEndpoint := by
1395  intro D
1396  let C :=
1397    TensorShearSector.EncodedTTHessianLichnerowiczCoeffOriginColumnFormulaData5.ofCoeffTranslatedData D
1398  let R :=
1399    TensorShearSector.EncodedTTHessianLichnerowiczRawOriginColumnFormulaData5.ofCoeffOriginColumnData C
1400  let O :=
1401    TensorShearSector.EncodedTTHessianLichnerowiczResidualOriginColumnTableFormulaData5.ofRawOriginColumnData R
1402  let T :=
1403    TensorShearSector.EncodedTTHessianLichnerowiczResidualOriginRowTableFormulaData5.ofRawOriginColumnData R
1404  let K :=
1405    TensorShearSector.EncodedTTHessianLichnerowiczResidualKernelFormulaData5.ofRawOriginColumnData R
1406  let E :=
1407    TensorShearSector.EncodedTTHessianLichnerowiczResidualEntryFormulaData5.ofRawOriginColumnData R
1408  let F :=
1409    TensorShearSector.PeriodicTTHessianLichnerowiczResidualEntryFormulaData5.ofRawOriginColumnData R
1410  let M :=
1411    TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5.ofCoeffTranslatedData D
1412  exact ⟨
1413    ⟨C⟩,
1414    ⟨R⟩,
1415    ⟨O⟩,
1416    ⟨T⟩,
1417    ⟨K⟩,
1418    ⟨E⟩,
1419    ⟨F⟩,
1420    ⟨M⟩,
1421    track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds⟩
1422
1423/-- Track 1.D diagnostic endpoint: a relative-frame translated coefficient
1424certificate at least exposes the origin-column consequence shared with the
1425absolute translated route.  This is intentionally weaker than the full-chain
1426absolute endpoint because physical stencil covariance still has to be converted
1427into the absolute row formula or into a shifted-generator theorem. -/
1428def Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedDiagnosticEndpoint : Prop :=
1429  TensorShearSector.EncodedTTHessianLichnerowiczCoeffRelativeTranslatedFormulaData5 →
1430    Nonempty TensorShearSector.EncodedTTHessianLichnerowiczCoeffRelativeOriginColumnFormulaData5
1431
1432/-- Relative-frame translated diagnostic endpoint consumed by Track 7. -/
1433theorem track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_diagnostic_endpoint_holds :
1434    Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedDiagnosticEndpoint := by
1435  intro D
1436  exact ⟨
1437    TensorShearSector.EncodedTTHessianLichnerowiczCoeffRelativeOriginColumnFormulaData5.ofCoeffRelativeTranslatedData D⟩
1438
1439/-- Track 1.D conditional endpoint: a relative-frame translated certificate
1440closes residual-zero on TT once the shifted-generator orthogonality lemma is
1441proved.  This names the exact remaining bridge from physical translation
1442covariance to the TT Hessian/Lichnerowicz route. -/
1443def Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedTTZeroEndpoint : Prop :=
1444  TensorShearSector.EncodedTTHessianLichnerowiczCoeffRelativeTranslatedTTZeroData5 →
1445    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5
1446
1447/-- Relative-frame translated conditional TT-zero endpoint consumed by Track 7. -/
1448theorem track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_ttzero_endpoint_holds :
1449    Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedTTZeroEndpoint := by
1450  intro D
1451  exact ⟨
1452    TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5.ofCoeffRelativeTranslatedTTZeroData D⟩
1453
1454/-- Track 1.D sharper conditional endpoint: generator-closure for every
1455relative row-frame translate closes the relative-frame residual-zero route. -/
1456def Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedClosureEndpoint : Prop :=
1457  TensorShearSector.EncodedTTHessianLichnerowiczCoeffRelativeTranslatedClosureData5 →
1458    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5
1459
1460/-- Relative-frame translated closure endpoint consumed by Track 7. -/
1461theorem track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_closure_endpoint_holds :
1462    Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedClosureEndpoint := by
1463  intro D
1464  exact ⟨
1465    TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5.ofCoeffRelativeTranslatedClosureData D⟩
1466
1467/-- Track 1.D closed relative endpoint: after proving shifted-generator closure,
1468a relative-frame translated coefficient certificate directly supplies
1469residual-zero on TT. -/
1470def Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedClosedTTZeroEndpoint : Prop :=
1471  TensorShearSector.EncodedTTHessianLichnerowiczCoeffRelativeTranslatedFormulaData5 →
1472    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5 ∧
1473    Nonempty TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5
1474
1475/-- Closed relative-frame translated TT-zero endpoint consumed by Track 7. -/
1476theorem track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_closed_ttzero_endpoint_holds :
1477    Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedClosedTTZeroEndpoint := by
1478  intro D
1479  exact ⟨
1480    ⟨TensorShearSector.PeriodicTTHessianLichnerowiczKernelResidualTTZeroData5.ofCoeffRelativeTranslatedData D⟩,
1481    ⟨TensorShearSector.PeriodicTTHessianLichnerowiczMatchData5.ofCoeffRelativeTranslatedData D⟩⟩
1482
1483/-- Session 210 Track 1.B local translation endpoint: the selected periodic
1484cube cell used by the explicit-fiber coefficient model commutes with `N=5`
1485torus translation. -/
1486def Track1MixedAxisSelectedCellTranslationEndpoint : Prop :=
1487  ∀ (a : Vertex5) (edge : PeriodicEdge5) (pair : FreudenthalLocalPair),
1488    selectedCell5 (translateEdge5 a edge) pair =
1489      translateVertex5 a (selectedCell5 edge pair)
1490
1491/-- Session 210 selected-cell translation endpoint consumed by Track 7. -/
1492theorem track1_mixed_axis_selected_cell_translation_endpoint_holds :
1493    Track1MixedAxisSelectedCellTranslationEndpoint :=
1494  selectedCell5_translate
1495
1496/-- Session 210 Track 1.B LHS row probe: the non-origin row at `(1,0,0)` is
1497translation-normalized on the mixed explicit-fiber LHS itself, independent of
1498the already-proved RHS stencil translation. -/
1499def Track1MixedAxisLhsRow100TranslationEndpoint : Prop :=
1500  rowMixedAxisLhsCoeffTranslationInvariant (1, 0, 0) = true
1501
1502/-- Session 210 LHS row-100 translation endpoint consumed by Track 7. -/
1503theorem track1_mixed_axis_lhs_row100_translation_endpoint_holds :
1504    Track1MixedAxisLhsRow100TranslationEndpoint :=
1505  rowMixedAxisLhsCoeffTranslationInvariant_100_eq_true
1506
1507/-- Session 210 Track 1.B local vertex translation endpoint: adding one of the
1508eight cube vertices after torus translation agrees with translating after the
1509local cube-vertex addition. -/
1510def Track1MixedAxisAddVertexBitsTranslationEndpoint : Prop :=
1511  ∀ (a cell : Vertex5) (b : Fin 8),
1512    Geometry.PeriodicFreudenthalTorus.addVertexBits (translateVertex5 a cell) b =
1513      translateVertex5 a (Geometry.PeriodicFreudenthalTorus.addVertexBits cell b)
1514
1515/-- Session 210 add-vertex-bits translation endpoint consumed by Track 7. -/
1516theorem track1_mixed_axis_add_vertex_bits_translation_endpoint_holds :
1517    Track1MixedAxisAddVertexBitsTranslationEndpoint :=
1518  addVertexBits_translate5
1519
1520/-- Session 210 Track 1.B edge-endpoint translation endpoint: translating a
1521periodic edge translates both endpoints and preserves its displacement. -/
1522def Track1MixedAxisEdgeEndpointsTranslationEndpoint : Prop :=
1523  ∀ (a : Vertex5) (edge : PeriodicEdge5),
1524    (translateEdge5 a edge).endpoints =
1525      (translateVertex5 a edge.endpoints.1, translateVertex5 a edge.endpoints.2)
1526
1527/-- Session 210 edge-endpoint translation endpoint consumed by Track 7. -/
1528theorem track1_mixed_axis_edge_endpoints_translation_endpoint_holds :
1529    Track1MixedAxisEdgeEndpointsTranslationEndpoint :=
1530  translateEdge5_endpoints
1531
1532/-- Fork B endpoint: local edge-stencil correspondence feeds the physical
1533finite-probe Regge/EH residual conclusion and the structural contracted
1534Bianchi interface.  This is an interface result; the manifold integral target
1535and concrete physical Schläfli identity remain open. -/
1536def Track1PhysicalResidualBianchiEndpoint : Prop :=
1537  ∀ (Nx Ny Nz : ℕ) [NeZero Nx] [NeZero Ny] [NeZero Nz]
1538    (hx : 2 < Nx) (hy : 2 < Ny) (hz : 2 < Nz)
1539    (V B : Type) [Fintype B],
1540    CanonicalPeriodicEdgeStencilLocalCorrespondence Nx Ny Nz hx hy hz →
1541      PhysicalReggeEHBianchiInterface.{0} Nx Ny Nz hx hy hz V B
1542
1543/-- Fork B endpoint theorem consumed by the integration lane. -/
1544theorem track1_physical_residual_bianchi_endpoint_holds :
1545    Track1PhysicalResidualBianchiEndpoint :=
1546  physicalReggeEHBianchiInterface_of_localCorrespondence
1547
1548/-- Agent B endpoint: once a concrete six-tet product-filter refinement family
1549is supplied, every slice has the concrete finite EH/Dirichlet limit-weight
1550target and the product-filter full-Regge aggregate converges to the family's
1551continuum EH integral. -/
1552def Track1ConcreteRiemannSumEndpoint : Prop :=
1553  ∀ {α ρ : Type*} {l : Filter α}
1554    (D : CanonicalPeriodicTetSixTetVolumeQuadratureProductFilterData (α := α) (ρ := ρ) l),
1555    PhysicalReggeEHConcreteRefinementFamilySliceTarget D.family ∧
1556    PhysicalReggeEHConcreteProductFilterTarget D ∧
1557    Nonempty (PhysicalReggeEHConcreteRefinementFamilyTargetCert (α := α) (ρ := ρ) l)
1558
1559/-- Agent B endpoint theorem consumed by the integration lane. -/
1560theorem track1_concrete_riemann_sum_endpoint_holds :
1561    Track1ConcreteRiemannSumEndpoint :=
1562  physicalReggeEH_concrete_refinement_family_target_one_statement
1563
1564/-- Agent B endpoint: once concrete product-filter refinement data are supplied,
1565the master theorem's D2 input can be instantiated with the physical Regge/EH
1566`Tendsto` target instead of the older flat-substrate structural identity. -/
1567def Track1PhysicalD2MasterWitnessEndpoint : Prop :=
1568  ∀ {α ρ : Type*} {l : Filter α}
1569    (D : CanonicalPeriodicTetSixTetVolumeQuadratureProductFilterData (α := α) (ρ := ρ) l)
1570    (V B : Type) [Fintype B],
1571    Nonempty (PhysicalReggeEHD2MasterWitnessCert D V B)
1572
1573/-- Agent B physical D2 master-witness endpoint theorem consumed by Track 7. -/
1574theorem track1_physical_d2_master_witness_endpoint_holds :
1575    Track1PhysicalD2MasterWitnessEndpoint :=
1576  fun D V B => physicalReggeEHD2MasterWitnessCert_inhabited D V B
1577
1578/-- Agent B endpoint: every concrete six-tet quadrature slice supplies actual
1579single-slice product-filter data.  This closes the `PUnit` cardinality case of
1580`CanonicalPeriodicTetSixTetVolumeQuadratureProductFilterData`; the genuinely
1581varying-cardinality product-filter data remains the manifold-scale target. -/
1582def Track1SingleSliceProductFilterDataEndpoint : Prop :=
1583  ∀ {α : Type*} {l : Filter α}
1584    (S : CanonicalPeriodicTetSixTetVolumeQuadratureSlice l)
1585    (refinementFilter : Filter PUnit.{1}),
1586    PhysicalReggeEHConcreteRefinementFamilySliceTarget
1587      (S.toSingleSliceProductFilterData refinementFilter).family ∧
1588    PhysicalReggeEHConcreteProductFilterTarget
1589      (S.toSingleSliceProductFilterData refinementFilter)
1590
1591/-- Agent B single-slice product-filter endpoint theorem consumed by Track 7. -/
1592theorem track1_single_slice_product_filter_data_endpoint_holds :
1593    Track1SingleSliceProductFilterDataEndpoint :=
1594  physicalReggeEH_concrete_single_slice_product_filter_data_one_statement
1595
1596/-- Agent B endpoint: staged cross-cardinality data plus a global residual
1597envelope supplies genuine varying-cardinality product-filter data.  This is the
1598non-`PUnit` route for the manifold-scale 1B-PHY target. -/
1599def Track1VaryingCardinalityProductFilterDataEndpoint : Prop :=
1600  ∀ {α ρ : Type*} {l : Filter α}
1601    {D : CanonicalPeriodicTetSixTetVolumeQuadratureCrossCardinalityData
1602      (α := α) (ρ := ρ) l}
1603    (E : CanonicalPeriodicTetSixTetVolumeQuadratureGlobalResidualEnvelopeData D),
1604    PhysicalReggeEHConcreteRefinementFamilySliceTarget E.toProductFilterData.family ∧
1605    PhysicalReggeEHConcreteProductFilterTarget E.toProductFilterData ∧
1606    Nonempty (PhysicalReggeEHConcreteRefinementFamilyTargetCert (α := α) (ρ := ρ) l)
1607
1608/-- Agent B varying-cardinality product-filter endpoint theorem consumed by
1609Track 7. -/
1610theorem track1_varying_cardinality_product_filter_data_endpoint_holds :
1611    Track1VaryingCardinalityProductFilterDataEndpoint :=
1612  fun E =>
1613    physicalReggeEH_concrete_varying_cardinality_product_filter_data_one_statement E
1614
1615/-- Agent B endpoint: the global residual envelope exposes the finite
1616full-Regge-to-quadrature residual estimate needed before product-filter
1617convergence. -/
1618def Track1FiniteProductResidualEstimateEndpoint : Prop :=
1619  ∀ {α ρ : Type*} {l : Filter α}
1620    {D : CanonicalPeriodicTetSixTetVolumeQuadratureCrossCardinalityData
1621      (α := α) (ρ := ρ) l}
1622    (E : CanonicalPeriodicTetSixTetVolumeQuadratureGlobalResidualEnvelopeData D),
1623    PhysicalReggeEHFiniteProductResidualEstimateTarget E
1624
1625theorem track1_finite_product_residual_estimate_endpoint_holds :
1626    Track1FiniteProductResidualEstimateEndpoint :=
1627  fun E => physicalReggeEHFiniteProductResidualEstimateTarget_holds E
1628
1629/-- Agent B endpoint: the finite product residual estimate normalizes to the raw
1630product-filter continuum `Tendsto` statement. -/
1631def Track1ContinuumNormalizationFromResidualEndpoint : Prop :=
1632  ∀ {α ρ : Type*} {l : Filter α}
1633    {D : CanonicalPeriodicTetSixTetVolumeQuadratureCrossCardinalityData
1634      (α := α) (ρ := ρ) l}
1635    (E : CanonicalPeriodicTetSixTetVolumeQuadratureGlobalResidualEnvelopeData D),
1636    PhysicalReggeEHContinuumNormalizationFromResidualTarget E
1637
1638theorem track1_continuum_normalization_from_residual_endpoint_holds :
1639    Track1ContinuumNormalizationFromResidualEndpoint :=
1640  fun E => physicalReggeEHContinuumNormalizationFromResidualTarget_holds E
1641
1642/-- Fork D endpoint: finite emitted recognition ticks induce bulk/radiation
1643capacity transfer, conserve total capacity, and evaluate to the existing
1644Schmidt `min` Page curve at the tick-induced evaporation fraction. -/
1645def Track3TickCapacityEndpoint : Prop :=
1646  (∀ S_BH N n, radiationCapacityFromTicks S_BH N n =
1647    radiationCapacity S_BH (evaporationFractionFromTicks N n)) ∧
1648  (∀ S_BH N n, 0 < N → n ≤ N →
1649    bulkCapacityFromTicks S_BH N n =
1650      bulkCapacity S_BH (evaporationFractionFromTicks N n)) ∧
1651  (∀ S_BH N n, 0 < N → n ≤ N →
1652    bulkCapacityFromTicks S_BH N n +
1653      radiationCapacityFromTicks S_BH N n = S_BH) ∧
1654  (∀ S_BH N n, 0 < N →
1655    radiationCapacityFromTicks S_BH N (n + 1) -
1656      radiationCapacityFromTicks S_BH N n = S_BH / (N : ℝ)) ∧
1657  (∀ S_BH N n, 0 < N → n + 1 ≤ N →
1658    bulkCapacityFromTicks S_BH N n -
1659      bulkCapacityFromTicks S_BH N (n + 1) = S_BH / (N : ℝ)) ∧
1660  (∀ S_BH N n, 0 < N → n ≤ N →
1661    pageCurveFromLedgerTicks S_BH N n =
1662      pageCurveFromUnitarity S_BH (evaporationFractionFromTicks N n)) ∧
1663  (∀ S_BH N, 0 ≤ S_BH → 0 < N →
1664    pageCurveFromLedgerTicks S_BH N 0 = 0) ∧
1665  (∀ S_BH N, 0 ≤ S_BH → 0 < N →
1666    pageCurveFromLedgerTicks S_BH N N = 0) ∧
1667  (∀ S_BH N n, 0 < N → n ≤ N →
1668    evaporationFractionFromTicks N n = 1 / 2 →
1669      pageCurveFromLedgerTicks S_BH N n = S_BH / 2)
1670
1671/-- Fork D endpoint theorem consumed by the integration lane. -/
1672theorem track3_tick_capacity_endpoint_holds : Track3TickCapacityEndpoint :=
1673  ⟨radiationCapacityFromTicks_eq_radiationCapacity,
1674   bulkCapacityFromTicks_eq_bulkCapacity,
1675   tick_capacity_sum_invariant,
1676   radiationCapacityFromTicks_next,
1677   bulkCapacityFromTicks_next,
1678   pageCurveFromLedgerTicks_eq_pageCurveFromUnitarity,
1679   pageCurveFromLedgerTicks_at_zero,
1680   pageCurveFromLedgerTicks_at_full,
1681   pageCurveFromLedgerTicks_at_page_fraction⟩
1682
1683/-- Agent D endpoint: explicit `BulkLedger ⊗ HawkingRadiation` carrier,
1684reversible linear tick operator, iterated operator evolution, and an entropy
1685readout interface connected to the ledger-tick Page curve.  This remains a
1686structural interface, not master-clause readiness. -/
1687def Track3OperatorProcessEndpoint : Prop :=
1688  Nonempty (BulkRadiationLedger (Fin 1) (Fin 1)) ∧
1689  Nonempty (PageTickUnitary (Fin 1) (Fin 1)) ∧
1690  Nonempty (OperatorPageEntropyReadout (Fin 1) (Fin 1)) ∧
1691  (∀ (P : OperatorPageProcess (Fin 1) (Fin 1)) (n : ℕ),
1692    P.stateAtTick (n + 1) = P.unitaryTick.tick (P.stateAtTick n)) ∧
1693  (∀ P : OperatorPageEntropyReadout (Fin 1) (Fin 1),
1694    P.radiationEntropyAtTick 0 = 0) ∧
1695  (∀ P : OperatorPageEntropyReadout (Fin 1) (Fin 1),
1696    P.radiationEntropyAtTick P.totalTicks = 0)
1697
1698/-- Agent D endpoint theorem consumed by the integration lane. -/
1699theorem track3_operator_process_endpoint_holds : Track3OperatorProcessEndpoint :=
1700  operator_page_process_interface_one_statement
1701
1702/-- Fork E endpoint: the Track 4.C structural `w(z)` lane has named
1703`z = 0.5` and `z = 1.0` falsifier thresholds plus a formal separation theorem
1704for measurements closer to ΛCDM than the RS structural band. -/
1705def Track4DarkEnergyFalsifierEndpoint : Prop :=
1706  (falsifierThreshold redshift_half = phi_neg_44 / 2) ∧
1707  (falsifierThreshold redshift_one = phi_neg_44) ∧
1708  (∀ z : ℝ, 0 < z → ∀ w_measured : ℝ,
1709    |w_measured - w_LCDM_value| < falsifierThreshold z →
1710      w_measured ≠ w_RS_linear z) ∧
1711  Nonempty DarkEnergyWofZStructuralCert
1712
1713/-- Fork E endpoint theorem consumed by the integration lane. -/
1714theorem track4_dark_energy_falsifier_endpoint_holds :
1715    Track4DarkEnergyFalsifierEndpoint :=
1716  ⟨falsifierThreshold_at_redshift_half,
1717   falsifierThreshold_at_redshift_one,
1718   fun z hz _w hclose => measured_near_LCDM_not_RS_linear z hz hclose,
1719   darkEnergyWofZStructuralCert_inhabited⟩
1720
1721/-- Fork F endpoint: Track 6 has theorem-grade discriminator sectors, rival
1722row coverage, dataset attachments, likelihood/status records, and the guarded
1723GWTC-3 ringdown runner packaged in one certificate. -/
1724def Track6SensitivityEndpoint : Prop :=
1725  (theoremGradeDiscriminatorSectors = 3) ∧
1726  (rivalRowsCovered = 4) ∧
1727  (falsifierRowsWithDatasetAttachments = 10) ∧
1728  (rowsWithLikelihoodOrStatusRecords = 6) ∧
1729  (guardedRingdownFamilies = 3) ∧
1730  (guardedRingdownMappings = 2) ∧
1731  Nonempty Track6FalsifierSensitivityCert
1732
1733/-- Fork F endpoint theorem consumed by the integration lane. -/
1734theorem track6_sensitivity_endpoint_holds : Track6SensitivityEndpoint :=
1735  track6_falsifier_sensitivity_one_statement
1736
1737/-! ## §2. Integrated handoff certificate -/
1738
1739/-- Integration certificate for Forks A, B, C, D, E, and F.
1740
1741The structural master theorem still uses structural witnesses where the master
1742plan says it must.  The new Track 2 many-body endpoint and Track 6 sensitivity
1743package are consumed here as stronger handoff facts; the Track 1 result is a
1744reduction/interface package, not a closure of the open Schläfli leaves. -/
1745structure ForkHandoffIntegrationCert where
1746  track2_many_body : Track2ManyBodyEndpoint
1747  track2_many_body_cert : Nonempty ManyBodyPhysicalChannelAmplitudeLinearCert
1748  track1_schlaefli_reduction : Track1SchlaefliReductionEndpoint
1749  track1_disp0_base_vertex_reduction :
1750    Track1Disp0BaseVertexReductionEndpoint
1751  track1_disp0_stationary_reduction :
1752    Track1Disp0StationaryReductionEndpoint
1753  track1_disp_stationary_reduction :
1754    Track1DispStationaryReductionEndpoint
1755  track1_seven_stationarity :
1756    Track1SevenStationarityEndpoint
1757  track1_forall_disp_stationarity_packaging :
1758    Track1ForallDispStationarityPackagingEndpoint
1759  track1_forall_disp_stationarity :
1760    Track1ForallDispStationarityEndpoint
1761  track1_total_symmetry_stationarity_reduction :
1762    Track1TotalSymmetryStationarityReductionEndpoint
1763  track1_total_symmetry_stationarity :
1764    Track1TotalSymmetryStationarityEndpoint
1765  track1_conformal_schlaefli :
1766    Track1ConformalSchlaefliEndpoint
1767  track1_conformal_schlaefli_local_expansion :
1768    Track1ConformalSchlaefliLocalExpansionEndpoint
1769  track1_conformal_schlaefli_near_zero_expansion :
1770    Track1ConformalSchlaefliNearZeroExpansionEndpoint
1771  track1_conformal_schlaefli_near_zero_local_reduction :
1772    Track1ConformalSchlaefliNearZeroLocalReductionEndpoint
1773  track1_conformal_schlaefli_near_zero_chain_rule :
1774    Track1ConformalSchlaefliNearZeroChainRuleEndpoint
1775  track1_conformal_schlaefli_near_zero_closed_form :
1776    Track1ConformalSchlaefliNearZeroClosedFormEndpoint
1777  track1_conformal_schlaefli_near_zero_local :
1778    Track1ConformalSchlaefliNearZeroLocalEndpoint
1779  track1_conformal_schlaefli_near_zero_stationarity :
1780    Track1ConformalSchlaefliNearZeroStationarityEndpoint
1781  track1_local_correspondence_reduced_to_mixed_length :
1782    Track1LocalCorrespondenceReducedToMixedLengthEndpoint
1783  track1_mixed_length_audit_obstruction :
1784    Track1MixedLengthAuditObstructionEndpoint
1785  track1_mixed_axis_stencil_reduction :
1786    Track1MixedAxisStencilReductionEndpoint
1787  track1_mixed_axis_coeff_cert :
1788    Track1MixedAxisCoeffCertEndpoint
1789  track1_mixed_axis_row100_coeff_cert :
1790    Track1MixedAxisRow100CoeffCertEndpoint
1791  track1_mixed_axis_origin_prop_coeff_cert :
1792    Track1MixedAxisOriginPropCoeffCertEndpoint
1793  track1_mixed_axis_translation_reduction :
1794    Track1MixedAxisTranslationReductionEndpoint
1795  track1_mixed_axis_stencil_rhs_translation :
1796    Track1MixedAxisStencilRhsTranslationEndpoint
1797  track1_mixed_axis_lhs_translation_reduction :
1798    Track1MixedAxisLhsTranslationReductionEndpoint
1799  track1_mixed_axis_edge_lhs_translation_reduction :
1800    Track1MixedAxisEdgeLhsTranslationReductionEndpoint
1801  track1_mixed_axis_edge_lhs_translation :
1802    Track1MixedAxisEdgeLhsTranslationEndpoint
1803  track1_mixed_axis_lhs_translation :
1804    Track1MixedAxisLhsTranslationEndpoint
1805  track1_mixed_axis_full_residual_coeff_cert :
1806    Track1MixedAxisFullResidualCoeffCertEndpoint
1807  track1_mixed_axis_rhs_soundness :
1808    Track1MixedAxisRhsSoundnessEndpoint
1809  track1_mixed_axis_explicit_fiber_lhs_soundness :
1810    Track1MixedAxisExplicitFiberLhsSoundnessEndpoint
1811  track1_mixed_axis_explicit_fiber_axis_soundness :
1812    Track1MixedAxisExplicitFiberAxisSoundnessEndpoint
1813  track1_mixed_axis_explicit_fiber_axis_stencil_target :
1814    Track1MixedAxisExplicitFiberAxisStencilTargetEndpoint
1815  track1_mixed_axis_corrected_axis_stencil_target :
1816    Track1MixedAxisCorrectedAxisStencilTargetEndpoint
1817  track1_mixed_axis_coeff_soundness_to_explicit_fiber :
1818    Track1MixedAxisCoeffSoundnessToExplicitFiberEndpoint
1819  track1_mixed_axis_coeff_soundness_to_axis_stencil :
1820    Track1MixedAxisCoeffSoundnessToAxisStencilEndpoint
1821  track1D_tensor_shear_scaffold :
1822    Track1DTensorShearScaffoldIntegrationEndpoint
1823  track1D_tt_orthogonal_surface :
1824    Track1DTTOrthogonalSurfaceEndpoint
1825  track1D_tt_projector_data_reduction :
1826    Track1DTTProjectorDataReductionEndpoint
1827  track1D_tt_finite_generator_projector_reduction :
1828    Track1DTTFiniteGeneratorProjectorReductionEndpoint
1829  track1D_conformal_generator_span :
1830    Track1DConformalGeneratorSpanEndpoint
1831  track1D_tt_gauge_generator_projector_reduction :
1832    Track1DTTGaugeGeneratorProjectorReductionEndpoint
1833  track1D_tt_generator_map_projector_reduction :
1834    Track1DTTGeneratorMapProjectorReductionEndpoint
1835  track1D_tt_longitudinal_projector_reduction :
1836    Track1DTTLongitudinalProjectorReductionEndpoint
1837  track1D_tt_longitudinal_coefficient_projector_reduction :
1838    Track1DTTLongitudinalCoefficientProjectorReductionEndpoint
1839  track1D_tt_longitudinal_coefficient_solution_reduction :
1840    Track1DTTLongitudinalCoefficientSolutionReductionEndpoint
1841  track1D_tt_normal_equation_reduction :
1842    Track1DTTNormalEquationReductionEndpoint
1843  track1D_tt_gram_system_reduction :
1844    Track1DTTGramSystemReductionEndpoint
1845  track1D_tt_gram_load_solver_reduction :
1846    Track1DTTGramLoadSolverReductionEndpoint
1847  track1D_tt_gram_load_image_reduction :
1848    Track1DTTGramLoadImageReductionEndpoint
1849  track1D_tt_gram_kernel_criterion_reduction :
1850    Track1DTTGramKernelCriterionReductionEndpoint
1851  track1D_tt_gram_kernel_generator_map_zero_reduction :
1852    Track1DTTGramKernelGeneratorMapZeroReductionEndpoint
1853  track1D_tt_gram_range_criterion_reduction :
1854    Track1DTTGramRangeCriterionReductionEndpoint
1855  track1D_tt_gram_self_adjoint :
1856    Track1DTTGramSelfAdjointEndpoint
1857  track1D_tt_gram_range_closed :
1858    Track1DTTGramRangeClosedEndpoint
1859  track1D_tt_hessian_lichnerowicz_bilinear_reduction :
1860    Track1DTTHessianLichnerowiczBilinearReductionEndpoint
1861  track1D_tt_hessian_lichnerowicz_kernel_row_reduction :
1862    Track1DTTHessianLichnerowiczKernelRowReductionEndpoint
1863  track1D_tt_hessian_lichnerowicz_kernel_entry_reduction :
1864    Track1DTTHessianLichnerowiczKernelEntryReductionEndpoint
1865  track1D_tt_hessian_lichnerowicz_residual_tt_zero_reduction :
1866    Track1DTTHessianLichnerowiczResidualTTZeroReductionEndpoint
1867  track1D_tt_hessian_lichnerowicz_residual_row_span_reduction :
1868    Track1DTTHessianLichnerowiczResidualRowSpanReductionEndpoint
1869  track1D_tt_hessian_lichnerowicz_residual_row_coeff_reduction :
1870    Track1DTTHessianLichnerowiczResidualRowCoeffReductionEndpoint
1871  track1D_tt_hessian_lichnerowicz_residual_row_coeff_entry_reduction :
1872    Track1DTTHessianLichnerowiczResidualRowCoeffEntryReductionEndpoint
1873  track1D_tt_hessian_lichnerowicz_residual_entry_formula_reduction :
1874    Track1DTTHessianLichnerowiczResidualEntryFormulaReductionEndpoint
1875  track1D_tt_hessian_lichnerowicz_encoded_residual_entry_formula_reduction :
1876    Track1DTTHessianLichnerowiczEncodedResidualEntryFormulaReductionEndpoint
1877  track1D_tt_hessian_lichnerowicz_encoded_residual_kernel_formula_reduction :
1878    Track1DTTHessianLichnerowiczEncodedResidualKernelFormulaReductionEndpoint
1879  track1D_tt_hessian_lichnerowicz_encoded_residual_disp_row_formula_reduction :
1880    Track1DTTHessianLichnerowiczEncodedResidualDispRowFormulaReductionEndpoint
1881  track1D_tt_hessian_lichnerowicz_encoded_residual_origin_row_table_reduction :
1882    Track1DTTHessianLichnerowiczEncodedResidualOriginRowTableReductionEndpoint
1883  track1D_tt_hessian_lichnerowicz_encoded_residual_origin_column_table_reduction :
1884    Track1DTTHessianLichnerowiczEncodedResidualOriginColumnTableReductionEndpoint
1885  track1D_tt_hessian_lichnerowicz_encoded_raw_origin_column_reduction :
1886    Track1DTTHessianLichnerowiczEncodedRawOriginColumnReductionEndpoint
1887  track1D_tt_hessian_lichnerowicz_encoded_coeff_origin_column_reduction :
1888    Track1DTTHessianLichnerowiczEncodedCoeffOriginColumnReductionEndpoint
1889  track1D_tt_hessian_lichnerowicz_encoded_coeff_translated_reduction :
1890    Track1DTTHessianLichnerowiczEncodedCoeffTranslatedReductionEndpoint
1891  track1D_tt_hessian_lichnerowicz_encoded_coeff_translated_full_chain :
1892    Track1DTTHessianLichnerowiczEncodedCoeffTranslatedFullChainEndpoint
1893  track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_diagnostic :
1894    Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedDiagnosticEndpoint
1895  track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_ttzero :
1896    Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedTTZeroEndpoint
1897  track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_closure :
1898    Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedClosureEndpoint
1899  track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_closed_ttzero :
1900    Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedClosedTTZeroEndpoint
1901  track1_mixed_axis_selected_cell_translation :
1902    Track1MixedAxisSelectedCellTranslationEndpoint
1903  track1_mixed_axis_lhs_row100_translation :
1904    Track1MixedAxisLhsRow100TranslationEndpoint
1905  track1_mixed_axis_add_vertex_bits_translation :
1906    Track1MixedAxisAddVertexBitsTranslationEndpoint
1907  track1_mixed_axis_edge_endpoints_translation :
1908    Track1MixedAxisEdgeEndpointsTranslationEndpoint
1909  track1_physical_residual_bianchi :
1910    Track1PhysicalResidualBianchiEndpoint
1911  track1_concrete_riemann_sum :
1912    Track1ConcreteRiemannSumEndpoint
1913  track1_physical_d2_master_witness :
1914    Track1PhysicalD2MasterWitnessEndpoint
1915  track1_single_slice_product_filter_data :
1916    Track1SingleSliceProductFilterDataEndpoint
1917  track1_varying_cardinality_product_filter_data :
1918    Track1VaryingCardinalityProductFilterDataEndpoint
1919  track1_finite_product_residual_estimate :
1920    Track1FiniteProductResidualEstimateEndpoint
1921  track1_continuum_normalization_from_residual :
1922    Track1ContinuumNormalizationFromResidualEndpoint
1923  track3_tick_capacity : Track3TickCapacityEndpoint
1924  track3_operator_process : Track3OperatorProcessEndpoint
1925  track4_dark_energy_falsifier : Track4DarkEnergyFalsifierEndpoint
1926  track6_sensitivity : Track6SensitivityEndpoint
1927  structural_master_cert : Nonempty MasterTheoremStructuralCert
1928
1929/-- The integration-lane certificate instance. -/
1930noncomputable def forkHandoffIntegrationCert : ForkHandoffIntegrationCert where
1931  track2_many_body := track2_many_body_endpoint_holds
1932  track2_many_body_cert := manyBodyPhysicalChannelAmplitudeLinearCert_inhabited
1933  track1_schlaefli_reduction := track1_schlaefli_reduction_endpoint_holds
1934  track1_disp0_base_vertex_reduction :=
1935    track1_disp0_base_vertex_reduction_endpoint_holds
1936  track1_disp0_stationary_reduction :=
1937    track1_disp0_stationary_reduction_endpoint_holds
1938  track1_disp_stationary_reduction :=
1939    track1_disp_stationary_reduction_endpoint_holds
1940  track1_seven_stationarity :=
1941    track1_seven_stationarity_endpoint_holds
1942  track1_forall_disp_stationarity_packaging :=
1943    track1_forall_disp_stationarity_packaging_endpoint_holds
1944  track1_forall_disp_stationarity :=
1945    track1_forall_disp_stationarity_endpoint_holds
1946  track1_total_symmetry_stationarity_reduction :=
1947    track1_total_symmetry_stationarity_reduction_endpoint_holds
1948  track1_total_symmetry_stationarity :=
1949    track1_total_symmetry_stationarity_endpoint_holds
1950  track1_conformal_schlaefli :=
1951    track1_conformal_schlaefli_endpoint_holds
1952  track1_conformal_schlaefli_local_expansion :=
1953    track1_conformal_schlaefli_local_expansion_endpoint_holds
1954  track1_conformal_schlaefli_near_zero_expansion :=
1955    track1_conformal_schlaefli_near_zero_expansion_endpoint_holds
1956  track1_conformal_schlaefli_near_zero_local_reduction :=
1957    track1_conformal_schlaefli_near_zero_local_reduction_endpoint_holds
1958  track1_conformal_schlaefli_near_zero_chain_rule :=
1959    track1_conformal_schlaefli_near_zero_chain_rule_endpoint_holds
1960  track1_conformal_schlaefli_near_zero_closed_form :=
1961    track1_conformal_schlaefli_near_zero_closed_form_endpoint_holds
1962  track1_conformal_schlaefli_near_zero_local :=
1963    track1_conformal_schlaefli_near_zero_local_endpoint_holds
1964  track1_conformal_schlaefli_near_zero_stationarity :=
1965    track1_conformal_schlaefli_near_zero_stationarity_endpoint_holds
1966  track1_local_correspondence_reduced_to_mixed_length :=
1967    track1_local_correspondence_reduced_to_mixed_length_endpoint_holds
1968  track1_mixed_length_audit_obstruction :=
1969    track1_mixed_length_audit_obstruction_endpoint_holds
1970  track1_mixed_axis_stencil_reduction :=
1971    track1_mixed_axis_stencil_reduction_endpoint_holds
1972  track1_mixed_axis_coeff_cert :=
1973    track1_mixed_axis_coeff_cert_endpoint_holds
1974  track1_mixed_axis_row100_coeff_cert :=
1975    track1_mixed_axis_row100_coeff_cert_endpoint_holds
1976  track1_mixed_axis_origin_prop_coeff_cert :=
1977    track1_mixed_axis_origin_prop_coeff_cert_endpoint_holds
1978  track1_mixed_axis_translation_reduction :=
1979    track1_mixed_axis_translation_reduction_endpoint_holds
1980  track1_mixed_axis_stencil_rhs_translation :=
1981    track1_mixed_axis_stencil_rhs_translation_endpoint_holds
1982  track1_mixed_axis_lhs_translation_reduction :=
1983    track1_mixed_axis_lhs_translation_reduction_endpoint_holds
1984  track1_mixed_axis_edge_lhs_translation_reduction :=
1985    track1_mixed_axis_edge_lhs_translation_reduction_endpoint_holds
1986  track1_mixed_axis_edge_lhs_translation :=
1987    track1_mixed_axis_edge_lhs_translation_endpoint_holds
1988  track1_mixed_axis_lhs_translation :=
1989    track1_mixed_axis_lhs_translation_endpoint_holds
1990  track1_mixed_axis_full_residual_coeff_cert :=
1991    track1_mixed_axis_full_residual_coeff_cert_endpoint_holds
1992  track1_mixed_axis_rhs_soundness :=
1993    track1_mixed_axis_rhs_soundness_endpoint_holds
1994  track1_mixed_axis_explicit_fiber_lhs_soundness :=
1995    track1_mixed_axis_explicit_fiber_lhs_soundness_endpoint_holds
1996  track1_mixed_axis_explicit_fiber_axis_soundness :=
1997    track1_mixed_axis_explicit_fiber_axis_soundness_endpoint_holds
1998  track1_mixed_axis_explicit_fiber_axis_stencil_target :=
1999    track1_mixed_axis_explicit_fiber_axis_stencil_target_endpoint_holds
2000  track1_mixed_axis_corrected_axis_stencil_target :=
2001    track1_mixed_axis_corrected_axis_stencil_target_endpoint_holds
2002  track1_mixed_axis_coeff_soundness_to_explicit_fiber :=
2003    track1_mixed_axis_coeff_soundness_to_explicit_fiber_endpoint_holds
2004  track1_mixed_axis_coeff_soundness_to_axis_stencil :=
2005    track1_mixed_axis_coeff_soundness_to_axis_stencil_endpoint_holds
2006  track1D_tensor_shear_scaffold :=
2007    track1D_tensor_shear_scaffold_integration_endpoint_holds
2008  track1D_tt_orthogonal_surface :=
2009    track1D_tt_orthogonal_surface_endpoint_holds
2010  track1D_tt_projector_data_reduction :=
2011    track1D_tt_projector_data_reduction_endpoint_holds
2012  track1D_tt_finite_generator_projector_reduction :=
2013    track1D_tt_finite_generator_projector_reduction_endpoint_holds
2014  track1D_conformal_generator_span :=
2015    track1D_conformal_generator_span_endpoint_holds
2016  track1D_tt_gauge_generator_projector_reduction :=
2017    track1D_tt_gauge_generator_projector_reduction_endpoint_holds
2018  track1D_tt_generator_map_projector_reduction :=
2019    track1D_tt_generator_map_projector_reduction_endpoint_holds
2020  track1D_tt_longitudinal_projector_reduction :=
2021    track1D_tt_longitudinal_projector_reduction_endpoint_holds
2022  track1D_tt_longitudinal_coefficient_projector_reduction :=
2023    track1D_tt_longitudinal_coefficient_projector_reduction_endpoint_holds
2024  track1D_tt_longitudinal_coefficient_solution_reduction :=
2025    track1D_tt_longitudinal_coefficient_solution_reduction_endpoint_holds
2026  track1D_tt_normal_equation_reduction :=
2027    track1D_tt_normal_equation_reduction_endpoint_holds
2028  track1D_tt_gram_system_reduction :=
2029    track1D_tt_gram_system_reduction_endpoint_holds
2030  track1D_tt_gram_load_solver_reduction :=
2031    track1D_tt_gram_load_solver_reduction_endpoint_holds
2032  track1D_tt_gram_load_image_reduction :=
2033    track1D_tt_gram_load_image_reduction_endpoint_holds
2034  track1D_tt_gram_kernel_criterion_reduction :=
2035    track1D_tt_gram_kernel_criterion_reduction_endpoint_holds
2036  track1D_tt_gram_kernel_generator_map_zero_reduction :=
2037    track1D_tt_gram_kernel_generator_map_zero_reduction_endpoint_holds
2038  track1D_tt_gram_range_criterion_reduction :=
2039    track1D_tt_gram_range_criterion_reduction_endpoint_holds
2040  track1D_tt_gram_self_adjoint :=
2041    track1D_tt_gram_self_adjoint_endpoint_holds
2042  track1D_tt_gram_range_closed :=
2043    track1D_tt_gram_range_closed_endpoint_holds
2044  track1D_tt_hessian_lichnerowicz_bilinear_reduction :=
2045    track1D_tt_hessian_lichnerowicz_bilinear_reduction_endpoint_holds
2046  track1D_tt_hessian_lichnerowicz_kernel_row_reduction :=
2047    track1D_tt_hessian_lichnerowicz_kernel_row_reduction_endpoint_holds
2048  track1D_tt_hessian_lichnerowicz_kernel_entry_reduction :=
2049    track1D_tt_hessian_lichnerowicz_kernel_entry_reduction_endpoint_holds
2050  track1D_tt_hessian_lichnerowicz_residual_tt_zero_reduction :=
2051    track1D_tt_hessian_lichnerowicz_residual_tt_zero_reduction_endpoint_holds
2052  track1D_tt_hessian_lichnerowicz_residual_row_span_reduction :=
2053    track1D_tt_hessian_lichnerowicz_residual_row_span_reduction_endpoint_holds
2054  track1D_tt_hessian_lichnerowicz_residual_row_coeff_reduction :=
2055    track1D_tt_hessian_lichnerowicz_residual_row_coeff_reduction_endpoint_holds
2056  track1D_tt_hessian_lichnerowicz_residual_row_coeff_entry_reduction :=
2057    track1D_tt_hessian_lichnerowicz_residual_row_coeff_entry_reduction_endpoint_holds
2058  track1D_tt_hessian_lichnerowicz_residual_entry_formula_reduction :=
2059    track1D_tt_hessian_lichnerowicz_residual_entry_formula_reduction_endpoint_holds
2060  track1D_tt_hessian_lichnerowicz_encoded_residual_entry_formula_reduction :=
2061    track1D_tt_hessian_lichnerowicz_encoded_residual_entry_formula_reduction_endpoint_holds
2062  track1D_tt_hessian_lichnerowicz_encoded_residual_kernel_formula_reduction :=
2063    track1D_tt_hessian_lichnerowicz_encoded_residual_kernel_formula_reduction_endpoint_holds
2064  track1D_tt_hessian_lichnerowicz_encoded_residual_disp_row_formula_reduction :=
2065    track1D_tt_hessian_lichnerowicz_encoded_residual_disp_row_formula_reduction_endpoint_holds
2066  track1D_tt_hessian_lichnerowicz_encoded_residual_origin_row_table_reduction :=
2067    track1D_tt_hessian_lichnerowicz_encoded_residual_origin_row_table_reduction_endpoint_holds
2068  track1D_tt_hessian_lichnerowicz_encoded_residual_origin_column_table_reduction :=
2069    track1D_tt_hessian_lichnerowicz_encoded_residual_origin_column_table_reduction_endpoint_holds
2070  track1D_tt_hessian_lichnerowicz_encoded_raw_origin_column_reduction :=
2071    track1D_tt_hessian_lichnerowicz_encoded_raw_origin_column_reduction_endpoint_holds
2072  track1D_tt_hessian_lichnerowicz_encoded_coeff_origin_column_reduction :=
2073    track1D_tt_hessian_lichnerowicz_encoded_coeff_origin_column_reduction_endpoint_holds
2074  track1D_tt_hessian_lichnerowicz_encoded_coeff_translated_reduction :=
2075    track1D_tt_hessian_lichnerowicz_encoded_coeff_translated_reduction_endpoint_holds
2076  track1D_tt_hessian_lichnerowicz_encoded_coeff_translated_full_chain :=
2077    track1D_tt_hessian_lichnerowicz_encoded_coeff_translated_full_chain_endpoint_holds
2078  track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_diagnostic :=
2079    track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_diagnostic_endpoint_holds
2080  track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_ttzero :=
2081    track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_ttzero_endpoint_holds
2082  track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_closure :=
2083    track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_closure_endpoint_holds
2084  track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_closed_ttzero :=
2085    track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_closed_ttzero_endpoint_holds
2086  track1_mixed_axis_selected_cell_translation :=
2087    track1_mixed_axis_selected_cell_translation_endpoint_holds
2088  track1_mixed_axis_lhs_row100_translation :=
2089    track1_mixed_axis_lhs_row100_translation_endpoint_holds
2090  track1_mixed_axis_add_vertex_bits_translation :=
2091    track1_mixed_axis_add_vertex_bits_translation_endpoint_holds
2092  track1_mixed_axis_edge_endpoints_translation :=
2093    track1_mixed_axis_edge_endpoints_translation_endpoint_holds
2094  track1_physical_residual_bianchi :=
2095    track1_physical_residual_bianchi_endpoint_holds
2096  track1_concrete_riemann_sum :=
2097    track1_concrete_riemann_sum_endpoint_holds
2098  track1_physical_d2_master_witness :=
2099    track1_physical_d2_master_witness_endpoint_holds
2100  track1_single_slice_product_filter_data :=
2101    track1_single_slice_product_filter_data_endpoint_holds
2102  track1_varying_cardinality_product_filter_data :=
2103    track1_varying_cardinality_product_filter_data_endpoint_holds
2104  track1_finite_product_residual_estimate :=
2105    track1_finite_product_residual_estimate_endpoint_holds
2106  track1_continuum_normalization_from_residual :=
2107    track1_continuum_normalization_from_residual_endpoint_holds
2108  track3_tick_capacity := track3_tick_capacity_endpoint_holds
2109  track3_operator_process := track3_operator_process_endpoint_holds
2110  track4_dark_energy_falsifier := track4_dark_energy_falsifier_endpoint_holds
2111  track6_sensitivity := track6_sensitivity_endpoint_holds
2112  structural_master_cert := masterTheoremStructuralCert_inhabited
2113
2114theorem forkHandoffIntegrationCert_inhabited :
2115    Nonempty ForkHandoffIntegrationCert :=
2116  ⟨forkHandoffIntegrationCert⟩
2117
2118/-- Session 565 projection: the integration certificate exposes the direct
2119uniform displacement-stationarity endpoint for Track 1.B-SCH. -/
2120theorem forkHandoffIntegrationCert_track1_forall_disp_stationarity :
2121    Track1ForallDispStationarityEndpoint :=
2122  track1_forall_disp_stationarity_endpoint_holds
2123
2124/-- Session 576 projection: the integration certificate exposes the
2125total-plus-symmetry reduction endpoint for Track 1.B-SCH. -/
2126theorem forkHandoffIntegrationCert_track1_total_symmetry_stationarity_reduction :
2127    Track1TotalSymmetryStationarityReductionEndpoint :=
2128  track1_total_symmetry_stationarity_reduction_endpoint_holds
2129
2130/-- Session 574 projection: the integration certificate exposes the direct
2131total-plus-symmetry stationarity endpoint for Track 1.B-SCH. -/
2132theorem forkHandoffIntegrationCert_track1_total_symmetry_stationarity :
2133    Track1TotalSymmetryStationarityEndpoint :=
2134  track1_total_symmetry_stationarity_endpoint_holds
2135
2136/-- **FORK A/B/C/D/E/F INTEGRATION ONE-STATEMENT.** Track 7 can now consume:
2137Fork C's many-body `PiTensorProduct` channel lift, Fork A's seven-leaf
2138Schläfli-to-stationarity reduction, Fork B's physical residual/Bianchi
2139interface, Fork D's tick-capacity Page layer, Fork E's `w(z)` falsifier bands,
2140Fork F's falsifier-sensitivity package, and the existing structural master
2141certificate.  This statement deliberately does not assert the fully
2142unconditional discovery theorem. -/
2143theorem fork_A_B_C_D_E_F_handoffs_integrated_one_statement :
2144    Track2ManyBodyEndpoint ∧
2145    Nonempty ManyBodyPhysicalChannelAmplitudeLinearCert ∧
2146    Track1SchlaefliReductionEndpoint ∧
2147    Track1Disp0BaseVertexReductionEndpoint ∧
2148    Track1Disp0StationaryReductionEndpoint ∧
2149    Track1DispStationaryReductionEndpoint ∧
2150    Track1SevenStationarityEndpoint ∧
2151    Track1ForallDispStationarityPackagingEndpoint ∧
2152    Track1ForallDispStationarityEndpoint ∧
2153    Track1TotalSymmetryStationarityReductionEndpoint ∧
2154    Track1TotalSymmetryStationarityEndpoint ∧
2155    Track1ConformalSchlaefliEndpoint ∧
2156    Track1ConformalSchlaefliLocalExpansionEndpoint ∧
2157    Track1ConformalSchlaefliNearZeroExpansionEndpoint ∧
2158    Track1ConformalSchlaefliNearZeroLocalReductionEndpoint ∧
2159    Track1ConformalSchlaefliNearZeroChainRuleEndpoint ∧
2160    Track1ConformalSchlaefliNearZeroClosedFormEndpoint ∧
2161    Track1ConformalSchlaefliNearZeroLocalEndpoint ∧
2162    Track1ConformalSchlaefliNearZeroStationarityEndpoint ∧
2163    Track1LocalCorrespondenceReducedToMixedLengthEndpoint ∧
2164    Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedDiagnosticEndpoint ∧
2165    Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedTTZeroEndpoint ∧
2166    Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedClosureEndpoint ∧
2167    Track1DTTHessianLichnerowiczEncodedCoeffRelativeTranslatedClosedTTZeroEndpoint ∧
2168    Track1PhysicalResidualBianchiEndpoint ∧
2169    Track1ConcreteRiemannSumEndpoint ∧
2170    Track1PhysicalD2MasterWitnessEndpoint ∧
2171    Track1SingleSliceProductFilterDataEndpoint ∧
2172    Track1VaryingCardinalityProductFilterDataEndpoint ∧
2173    Track1FiniteProductResidualEstimateEndpoint ∧
2174    Track1ContinuumNormalizationFromResidualEndpoint ∧
2175    Track3TickCapacityEndpoint ∧
2176    Track3OperatorProcessEndpoint ∧
2177    Track4DarkEnergyFalsifierEndpoint ∧
2178    Track6SensitivityEndpoint ∧
2179    Nonempty MasterTheoremStructuralCert :=
2180  ⟨track2_many_body_endpoint_holds,
2181   manyBodyPhysicalChannelAmplitudeLinearCert_inhabited,
2182   track1_schlaefli_reduction_endpoint_holds,
2183   track1_disp0_base_vertex_reduction_endpoint_holds,
2184   track1_disp0_stationary_reduction_endpoint_holds,
2185   track1_disp_stationary_reduction_endpoint_holds,
2186   track1_seven_stationarity_endpoint_holds,
2187   track1_forall_disp_stationarity_packaging_endpoint_holds,
2188   track1_forall_disp_stationarity_endpoint_holds,
2189   track1_total_symmetry_stationarity_reduction_endpoint_holds,
2190   track1_total_symmetry_stationarity_endpoint_holds,
2191   track1_conformal_schlaefli_endpoint_holds,
2192   track1_conformal_schlaefli_local_expansion_endpoint_holds,
2193   track1_conformal_schlaefli_near_zero_expansion_endpoint_holds,
2194   track1_conformal_schlaefli_near_zero_local_reduction_endpoint_holds,
2195   track1_conformal_schlaefli_near_zero_chain_rule_endpoint_holds,
2196   track1_conformal_schlaefli_near_zero_closed_form_endpoint_holds,
2197   track1_conformal_schlaefli_near_zero_local_endpoint_holds,
2198   track1_conformal_schlaefli_near_zero_stationarity_endpoint_holds,
2199   track1_local_correspondence_reduced_to_mixed_length_endpoint_holds,
2200   track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_diagnostic_endpoint_holds,
2201   track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_ttzero_endpoint_holds,
2202   track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_closure_endpoint_holds,
2203   track1D_tt_hessian_lichnerowicz_encoded_coeff_relative_translated_closed_ttzero_endpoint_holds,
2204   track1_physical_residual_bianchi_endpoint_holds,
2205   track1_concrete_riemann_sum_endpoint_holds,
2206   track1_physical_d2_master_witness_endpoint_holds,
2207   track1_single_slice_product_filter_data_endpoint_holds,
2208   track1_varying_cardinality_product_filter_data_endpoint_holds,
2209   track1_finite_product_residual_estimate_endpoint_holds,
2210   track1_continuum_normalization_from_residual_endpoint_holds,
2211   track3_tick_capacity_endpoint_holds,
2212   track3_operator_process_endpoint_holds,
2213   track4_dark_energy_falsifier_endpoint_holds,
2214   track6_sensitivity_endpoint_holds,
2215   masterTheoremStructuralCert_inhabited⟩
2216
2217/-- Session 565 projection: the integrated Fork A/B/C/D/E/F one-statement exposes
2218the direct uniform displacement-stationarity endpoint. -/
2219theorem fork_A_B_C_D_E_F_handoffs_integrated_one_statement_track1_forall_disp_stationarity :
2220    Track1ForallDispStationarityEndpoint :=
2221  track1_forall_disp_stationarity_endpoint_holds
2222
2223/-- Session 576 projection: the integrated Fork A/B/C/D/E/F one-statement exposes
2224the total-plus-symmetry reduction endpoint. -/
2225theorem fork_A_B_C_D_E_F_handoffs_integrated_one_statement_track1_total_symmetry_stationarity_reduction :
2226    Track1TotalSymmetryStationarityReductionEndpoint :=
2227  track1_total_symmetry_stationarity_reduction_endpoint_holds
2228
2229/-- Session 574 projection: the integrated Fork A/B/C/D/E/F one-statement exposes
2230the direct total-plus-symmetry stationarity endpoint. -/
2231theorem fork_A_B_C_D_E_F_handoffs_integrated_one_statement_track1_total_symmetry_stationarity :
2232    Track1TotalSymmetryStationarityEndpoint :=
2233  track1_total_symmetry_stationarity_endpoint_holds
2234
2235/-- Session 565 audit count for the direct uniform-stationarity handoff accessors:
2236one certificate field projection and one integrated one-statement projection. -/
2237def track1ForallDispStationarityHandoffProjectionCount : ℕ := 2
2238
2239theorem track1ForallDispStationarityHandoffProjectionCount_eq_two :
2240    track1ForallDispStationarityHandoffProjectionCount = 2 := rfl
2241
2242/-- Session 576 audit count for the total-plus-symmetry reduction handoff
2243accessors: one certificate field projection and one integrated one-statement
2244projection. -/
2245def track1TotalSymmetryStationarityReductionHandoffProjectionCount : ℕ := 2
2246
2247theorem track1TotalSymmetryStationarityReductionHandoffProjectionCount_eq_two :
2248    track1TotalSymmetryStationarityReductionHandoffProjectionCount = 2 := rfl
2249
2250/-- Session 574 audit count for the direct total-plus-symmetry stationarity
2251handoff accessors: one certificate field projection and one integrated
2252one-statement projection. -/
2253def track1TotalSymmetryStationarityHandoffProjectionCount : ℕ := 2
2254
2255theorem track1TotalSymmetryStationarityHandoffProjectionCount_eq_two :
2256    track1TotalSymmetryStationarityHandoffProjectionCount = 2 := rfl
2257
2258/-- Backward-compatible name for the first integration receipt. -/
2259theorem fork_A_C_F_handoffs_integrated_one_statement :
2260    Track2ManyBodyEndpoint ∧
2261    Nonempty ManyBodyPhysicalChannelAmplitudeLinearCert ∧
2262    Track1SchlaefliReductionEndpoint ∧
2263    Track6SensitivityEndpoint ∧
2264    Nonempty MasterTheoremStructuralCert :=
2265  ⟨track2_many_body_endpoint_holds,
2266   manyBodyPhysicalChannelAmplitudeLinearCert_inhabited,
2267   track1_schlaefli_reduction_endpoint_holds,
2268   track6_sensitivity_endpoint_holds,
2269   masterTheoremStructuralCert_inhabited⟩
2270
2271end MasterTheoremHandoffIntegration
2272end Gravity
2273end IndisputableMonolith
2274